Symplectic connections

Pierre Bieliavsky1

Michel Cahen2

Simone Gutt2,3

John Rawnsley4

Lorenz Schwachhofer5

Abstract

This article is an overview of the results obtained in recent years on symplectic connections. We

present what is known about preferred connections (critical points of a variational principle). The

class of Ricci-type connections (for which the curvature is entirely determined by the Ricci tensor)

is described in detail, as well as its far reaching generalization to special connections. A twistorial

construction shows a relation between Ricci-type connections and complex geometry. We give a

construction of Ricci-flat symplectic connections. We end up by presenting, through an explicit

example, an approach to noncommutative symplectic symmetric spaces.

math.SG/0511194 v2, May 2006. Section 6.8 rewritten.

1Univ. Cath. Louvain, Dept de Math, ch. du cyclotron 2, B-1348 Louvain-la-Neuve, Belgium2Universite Libre de Bruxelles, Campus Plaine, CP 218, B-1050 Brussels, Belgium3Universite de Metz, Dept. de Math., Ile du Saulcy, F-57045 Metz Cedex 01, France4Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom5Math. Institut, Universitat Dortmund, Vogelpothsweg 87, D-44221 Dortmund, Germany

bieliavsky@math.ucl.ac.be, mcahen@ulb.ac.be, sgutt@ulb.ac.be,

j.rawnsley@warwick.ac.uk, lschwach@math.uni-dortmund.de

Contents

1 Introduction 1

2 Definitions and basic facts about symplectic connections 1

2.1 Existence and non-uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Where do symplectic connections arise? . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.3 When is there a “natural” unique symplectic connection? . . . . . . . . . . . . . . . . . 32.4 Curvature tensor of a symplectic connection . . . . . . . . . . . . . . . . . . . . . . . . . 42.5 Variational principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Preferred symplectic connections 6

3.1 Preferred symplectic connections in dimension 2 . . . . . . . . . . . . . . . . . . . . . . 73.2 Ricci-flat connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3 Ricci-parallel symplectic connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.4 Homogeneous preferred connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4 Ricci-type connections 11

4.1 Some properties of the curvature of a Ricci-type connection . . . . . . . . . . . . . . . . 114.2 A construction by reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.3 Local models for Ricci-type connections . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.4 Global models for Ricci-type connections . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5 Special symplectic connections 14

6 Symplectic twistor space and Ricci-type connections 19

6.1 Compatible almost complex structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.2 Geometry of j(V,Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.3 The bundle J(M,ω) of almost complex structures . . . . . . . . . . . . . . . . . . . . . . 206.4 Almost symplectic connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216.5 Differential geometry on J(M,ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.6 The almost complex structures on J(M,ω) . . . . . . . . . . . . . . . . . . . . . . . . . 236.7 Generalisations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.8 The bundle J(M,ω) for spaces of Ricci-type . . . . . . . . . . . . . . . . . . . . . . . . . 25

7 Ricci-flat connections 26

7.1 A construction by induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.2 Examples of contact quadruples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297.3 More about reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

8 Non-commutative symplectic symmetric spaces 32

8.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328.2 Basic definitions and the cocyclic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.3 A curved example: SO(1, 1)× R2/R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

References 39

i

1 Introduction

Symplectic geometry is by nature non-local. This is emphasized in particular by the classical Darbouxtheorem. The introduction on a given symplectic manifold of a symplectic connection, which is atool adapted to local computations, may seem inappropriate. The aim of this survey is to show thatsymplectic geometry in the presence of special symplectic connections becomes highly rigid. Moreprecisely there exists a family of universal models such that any symplectic manifold admitting sucha special symplectic connection is locally symplectically and affinely equivalent to a particular modelof the family. This local rigidity becomes a global rigidity if one requires compactness and simpleconnectedness.

It also appears that a twistor bundle over some of these symplectic manifolds with special connectionsadmits a natural structure of complex analytic manifold.

Among the universal models there are certain symmetric symplectic spaces. These are particularmanifolds where quantisation (whether formal or convergent) may be performed explicitly; hence theyare a good framework for non-commutative geometry.

Although this survey is not exhaustive we have taken care to explain various criteria for choosingparticular symplectic connections. Construction of quantisation based on a choice of symplectic con-nection is reduced here to one example, but we believe this example shows possibilities for development.We hope that this overview may lead others to investigate the interplay of symplectic geometry andsymplectic connections.

2 Definitions and basic facts about symplectic connections

2.1 Existence and non-uniqueness

Definition 2.1 Let (M,ω) be a smooth symplectic manifold of dimension 2n (i.e. ω is a closed non-degenerate 2-form on M). A symplectic connection on (M,ω) is a smooth linear connection ∇ suchthat:

– its torsion T∇ vanishes(⇔ T∇(X, Y ) := ∇XY −∇Y X − [X, Y ] = 0);

– the symplectic form ω is parallel(⇔ (∇Xω)(Y, Z) := X(ω(Y, Z))− ω(∇XY, Z)− ω(Y,∇XZ) = 0).

To prove the existence of such a connection, take ∇0 any torsion free linear connection (for instance,the Levi Civita connection associated to a metric g on M). Consider the tensor N on M defined by

∇0Xω(Y, Z) =: ω(N(X, Y ), Z).

Since ω is skewsymmetric we have ω(N(X, Y ), Z) = −ω(N(X, Z), Y ) and since ω is closed we have+

XY Z

ω(N(X, Y ), Z) = 0 where +XY Z

denotes the sum over cyclic permutations of the indices X, Y and

Z. Define∇XY := ∇0

XY +13N(X, Y ) +

13N(Y, X).

1

Then ∇ is torsion free and:

∇Xω(Y, Z) = X(ω(Y, Z))− ω(∇XY, Z)− ω(Y,∇XZ)

= ∇0Xω(Y, Z)− 1

3ω(N(X, Y ), Z)− 1

3ω(N(Y, X), Z)

−13ω(Y, N(X, Z))− 1

3ω(Y, N(Z,X))

= ∇0Xω(Y, Z)− 1

3ω(N(X, Y ), Z)− 1

3ω(N(Y, X), Z)

−13ω(N(X, Y ), Z)− 1

3ω(N(Z, Y ), X)

= (1− 13− 1

3)ω(N(X, Y ), Z) +

13ω(N(X, Z), Y ) = 0

so the linear connection ∇ is symplectic.We shall now see how (non)-unique is a symplectic connection. Take ∇ symplectic; then ∇′XY :=

∇XY + S(X, Y ) is symplectic if and only if S(X, Y ) = S(Y, X) (torsion free) and

0 = ∇′Xω(Y, Z)

= ∇Xω(Y, Z)− ω(S(X, Y ), Z)− ω(Y, S(X, Z))

= −ω(S(X, Y ), Z) + ω(S(X, Z), Y ),

hence if and only if ω(S(X, Y ), Z) is totally symmetric.Summarising we can now state the well known result:

Theorem 2.2 On a symplectic manifold (M,ω) there always exist symplectic connections. The set ofsymplectic connections is an affine space modelled on the space of contravariant symmetric 3-tensorfields on M , Γ∞(S3TM).

2.2 Where do symplectic connections arise?

The notion of symplectic connection is intimately related to that of natural formal deformation quanti-sation at order 2. Quantisation of a classical system is a way to pass from classical to quantum results.Deformation quantisation was introduced by Flato, Lichnerowicz and Sternheimer in [26] and in [3];they

“suggest that quantisation be understood as a deformation of the structure of the algebra of classicalobservables rather than a radical change in the nature of the observables.”

So deformation quantisation is defined in terms of a star product which is a formal deformation ofthe algebraic structure of the space of smooth functions on a symplectic (or more generally a Poisson)manifold. The associative structure given by the usual product of functions and the Lie structure givenby the Poisson bracket are simultaneously deformed. Let us recall that if (M,ω) is a symplectic manifoldand if u, v ∈ C∞(M), the Poisson bracket of u and v is defined by

u, v := Xu(v) = ω(Xv, Xu),

where Xu denotes the Hamiltonian vector field corresponding to the function u, i.e. such that i(Xu)ω =du.

2

Definition 2.3 A star product on a symplectic manifold (M,ω) is a bilinear map

C∞(M)× C∞(M)→ C∞(M)[[ν]] (u, v) 7→ u ∗ν v :=∑r≥0

νrCr(u, v)

such that(u ∗ v) ∗ w = u ∗ (v ∗ w) (when extended R[[ν]] linearly);C0(u, v) = uv C1(u, v)− C1(v, u) = u, v;1 ∗ u = u ∗ 1 = u.If all the Cr’s are bidifferential operators; one speaks of a differential star product; if, furthermore,each Cr is of order ≤ r in each argument, one speaks of a natural star product.

The link between symplectic connections and star products appear already in the seminal paper [3]where the authors observe that if there is a flat symplectic connection ∇ on (M,ω), one can generalisethe classical formula for Moyal star product ∗M defined on R2n with a constant symplectic 2-form.

Fedosov, proved more generally that given any symplectic connection ∇, one can construct a starproduct (in [27] it was proposed that a triple (M,ω,∇) be known as a Fedosov manifold):

Theorem 2.4 [25] Given a symplectic connection ∇ and a sequence Ω =∑∞

k=1 νkωk of closed 2-forms on a symplectic manifold (M,ω), one can build a star product ∗∇,Ω on it. This is obtained byidentifying the space C∞(M)[[ν]] with a subalgebra of the algebra of sections of a bundle of associativealgebras (called the Weyl bundle) on M . The subalgebra is the one of flat sections of the Weyl bundle,when this bundle is endowed with a flat connection whose construction is determined by the choicesmade of the connection on M and of the sequence of closed 2-forms on M .

Reciprocally a natural star product determines a symplectic connection. This was first observed byLichnerowicz [33] for a restricted class of star products.

Theorem 2.5 [28] A natural star product at order 2 determines a unique symplectic connection.

2.3 When is there a “natural” unique symplectic connection?

To have a canonical choice of symplectic connection on (M,ω), one needs some extra structure on themanifold.• Example 1: pseudo-Kahler manifolds

Choose an almost complex structure J on (M,ω) [i.e. J : TM → TM is a bundle endomorphism sothat J2 = − Id] so that ω(JX, JY ) = ω(X, Y )].A symplectic connection ∇ preserves J [i.e. ∇J = 0] if and only if it is the Levi Civita connectionassociated to the pseudo Riemannian metric g(X, Y ) := ω(X, JY ). It is thus unique and it only existsin a (pseudo-)Kahler situation.• Example 2: symmetric symplectic spaces

Intuitively, a symmetric symplectic space is a symplectic manifold with symmetries attached to each ofits points. Precisely:

Definition 2.6 A symmetric symplectic space is a triple (M,ω, S) where (M,ω) is a symplecticmanifold and where S is a smooth map S : M ×M → M such that, defining for any point x ∈ M themap (called the symmetry at x):

sx := S(x, ·) : M →M,

3

each sx squares to the identity [s2x = Id] and is a symplectomorphism of (M,ω) [s∗xω = ω],

x is an isolated fixed point of sx, and sxsysx = ssxy for any x, y ∈M .

Proposition 2.7 [4] On a symmetric symplectic space, there is a unique symplectic connection forwhich each sx is an affinity. It is explicitly given by

ωx(∇XY, Z) =12Xxω(Y + sx?Y, Z).

• In the two examples above, the choice of symplectic connection was imposed by the presence ofan additional structure. To select a “small” class of symplectic connections on a symplectic manifoldwithout any additional structure, one has to choose some restrictive conditions. One way to proceed isto impose some system of equations on the curvature tensor.

2.4 Curvature tensor of a symplectic connection

The curvature tensor R∇ of a linear connection ∇ is the 2-form on M with values in the endomor-phisms of the tangent bundle defined by

R∇(X, Y )Z =(∇X∇Y −∇Y∇X −∇[X,Y ]

)Z (1)

for vector fields X, Y, Z on M . If ∇ is symplectic, R∇x (X, Y ) has values in the symplectic Lie algebra

sp(TxM,ωx) = A ∈ End(TxM) | ωx(Au, v) + ωx(u, Av) = 0, ∀u, v ∈ TxM.The curvature tensor satisfies the first Bianchi identity

+X,Y,Z

R∇(X, Y )Z = 0

where + denotes the sum over the cyclic permutations of the listed set of elements,and the secondBianchi identity

+X,Y,Z

(∇XR∇) (Y, Z) = 0.

The Ricci tensor r∇ is the 2-tensor

r∇(X, Y ) = Tr(Z 7→ R∇(X, Z)Y

). (2)

The first Bianchi identity implies that r∇ is symmetric.One can define a second trace r′x(X, Y ) :=

∑i ω(R∇

x (ei, ei)X, Y ) where the ei constitute a basis of

TxM and the ei constitute the dual basis of TxM (i.e. such that ω(ei, ej) = δj

i ). Then Bianchi’s firstidentity implies that r′ = −2r∇.

Since the Ricci tensor is symmetric and one only has a skewsymmetric contravariant 2-tensor on M

(the Poisson tensor related to the symplectic form) there is no “scalar curvature”.The symplectic curvature tensor is defined as

R∇(X, Y, Z, T ) := ω(R∇(X, Y )Z, T ). (3)

It is antisymmetric in its first two arguments and symmetric in its last two. Hence R∇x is in Λ2(T ∗x M)⊗

S2(T ∗x M). To understand Bianchi’s first identity, we introduce the operators of symmetrisation andskewsymmetrisation arising in the Koszul long exact sequence.

4

2.4.1 The Koszul long exact sequence

Given any finite dimensional vector space V , the Koszul long exact sequence has the following form:

0 −→ Sq(V ) a−→V ⊗ Sq−1(V ) a−→Λ2V ⊗ Sq−2(V ) a−→· · · a−→Λq−1(V )⊗ Va−→Λq(V ) −→ 0

where a is the skewsymmetrisation operator:

a(v1 ∧ . . . ∧ vq ⊗ w1 · · ·wp) =p∑

i=1

v1 ∧ . . . ∧ vq ∧ wi ⊗ w1 · · ·wi−1wi+1 · · ·wp.

The symmetrisation operator reads:

s(v1 ∧ . . . ∧ vq ⊗ w1 · · ·wp)q∑

i=1

(−1)q−iv1 ∧ . . . vi−1 ∧ vi+1 . . . ∧ vq ⊗ vi · w1 · · ·wp.

These two operators satisfy a2 = 0, s2 = 0, (as + sa)|ΛqV⊗Sp(V )= (p + q) Id .

The first Bianchi identity on the value at a point x of the symplectic curvature tensor takes theform:

+X,Y,Z

R∇x (X, Y, Z, T ) = 0⇔ R∇

x ∈ ker a ⊂ Λ2(T ∗x M)⊗ S2(T ∗x M).

The space Rx of 4-tensors satisfying the algebraic identities of a symplectic curvature tensor at x is:

Rx := ker a|Λ2(V )⊗S2(V )'(V ⊗ S3(V )

)/S4(V ) for V = T ∗x M.

2.4.2 Decomposition of the curvature

The group Sp(TxM,ωx) = A ∈ End(TxM) | ωx(Au, Av) = ωx(u, v) ∀u, v ∈ TxM acts on V = T ∗x M

and thus on Rx '(V ⊗ S3(V )

)/S4(V ). Under this action the space V ⊗ S3(V ), in dimension 2n ≥ 4,

decomposes into three irreducible subspaces (S4(V )⊕ S′2(V )⊕W where S′2(V ) = a(s(ωx ⊗ S2(V ))) ∼S2(V )so that:

Rx = E x ⊕W x.

The decomposition of the symplectic curvature tensor R∇x into its E x component (denoted E∇

x ) and itsW x component (denoted W∇

x ) ,R∇

x = E∇x + W∇

x ,

is given by

E∇x (X, Y, Z, T ) = − 1

2(n + 1)[2ωx(X, Y )r∇x (Z, T ) + ωx(X, Z)r∇x (Y, T )

+ωx(X, T )r∇x (Y, Z)− ωx(Y, Z)r∇x (X, T )− ωx(Y, T )r∇x (X, Z)].

The corresponding decomposition of the curvature tensor (see Vaisman [45]) has the form

R∇x = E∇

x + W∇x , (4)

where

E∇(X, Y )Z = 12n+2

(2ω(X, Y )ρ∇Z + ω(X, Z)ρ∇Y − ω(Y, Z)ρ∇X (5)

+ω(X, ρ∇Z)Y − ω(Y, ρ∇Z)X)

5

with r∇ converted into an endomorphism ρ∇ given by

ω(X, ρ∇Y ) = r∇(X, Y ). (6)

Definition 2.8 A symplectic connection ∇ on (M,ω) will be said to be of Ricci-type if W∇ = 0; itwill be said to be Ricci-flat if E∇ = 0 (hence if and only if r∇ = 0).

One can combine restricting the holonomy algebra g ⊂ sp(R2n,Ω) and the vanishing of some componentsof the curvature when the curvature is decomposed into irreducible components under the action of g

to define special symplectic connections; these will appear in section 5.

2.5 Variational principle

To select symplectic connections through a variational principle [13], one can consider a LagrangianL(R∇), which is a polynomial in the curvature of the connection ∇, invariant under the action of thesymplectic group ∫

M

L(R∇)ωn.

From what we have seen before, there is no invariant polynomial of degree 1 in the curvature, so theeasiest choice is a polynomial of degree 2 in R∇. The space of degree 2 polynomials in the curvaturewhich are invariant under the action of the symplectic group is 2-dimensional and spanned by E∇ ·E∇

and W∇ ·W∇ (or, equivalently by R∇ ·R∇ and r∇ · r∇) where · denotes the symmetric function-valuedproduct of tensors induced by ω and S · T , for S and T tensor-fields on M of the same type, is given inlocal coordinates by

S · T = (ω−1)i1i′1 · · · (ω−1)ipi′pωj1j′1· · ·ωjqj′qS

j1...jq

i1...ipT

j′1...j′qi′1...i′p

.

From Chern–Weil theory we know that the first Pontryagin class is represented by a 4-form P1(∇)which is built from an invariant combination of the curvature and the symplectic form. In fact

P1(∇) ∧ ωn−2 =1

16π2[r∇ · r∇ − 1

2R∇ ·R∇]ωn

Since this combination will be constant under variations, all non-trivial Euler equations coming from avariational principle built from a second order invariant polynomial in the curvature are the same andtake the form:

+X,Y,Z

(∇Xr∇)(Y, Z) = 0. (7)

Definition 2.9 A symplectic connection ∇ is said to be preferred if it is a solution of Equation (7).

3 Preferred symplectic connections

The preferred symplectic connections ∇ on a symplectic manifold (M,ω) of dimension 2n are criticalpoints of the functional ∫

M

Tr(ρ∇)2ωn

n!

6

where ρ∇ is the Ricci endomorphism as previously defined in (6). They obey the system of second orderpartial differential equations

+X,Y,Z

(∇Xr∇)(Y, Z) = 0

where r∇ is the Ricci tensor of the connection ∇.The basic problem is to determine if on a given manifold (M,ω) there exists a preferred connection and“how many” different ones may occur; two solutions are different in this context if they are not relatedthrough a symplectic diffeomorphism.

This basic problem is essentially solved when (M,ω) is a compact orientable surface. We give alsoa partial answer for the standard symplectic plane (R2,Ω0).

When the dimension of M is ≥ 4, we can give large classes of examples of preferred connections:(i) examples of symplectic manifolds admitting a Ricci-flat connection;(ii) examples of symplectic manifolds admitting a “Ricci-parallel” connection;(iii) examples of homogeneous symplectic manifolds admitting a homogeneous preferred symplecticconnection.

3.1 Preferred symplectic connections in dimension 2

The study of preferred connections in dimension 2 relies on a function β, depending on the connection,which first appears in

Lemma 3.1 [13] Let (M,ω) be a symplectic surface and let ∇ be a preferred symplectic connection onit. Then

(i) there exists a 1-form u such that, for any vector fields X, Y, Z one has

(∇Xr∇)(Y, Z) = ω(Y, X)u(Z) + ω(Z,X)u(Y ); (8)

(ii) there exists a function β such that∇Xu = βω; (9)

(iii) defining the vector field u byi(u)ω = u,

one has, for any vector fields X, Y :

Xβ = −r∇(X, u),(∇2β)(X, Y ) = −u(X)u(Y ) + βr∇(X, Y );

(iv) there exist two real numbers A and B such that

r∇(u, u) = β2 + B,14

Tr(ρ∇)2 = β + A.

Hence ∇ is locally symmetric (i.e. ∇R∇ = 0) if and only if β = 0. If M is compact, the secondproperty above shows that β can not be a non-vanishing constant (it would indeed imply that thesymplectic 2-form ω be exact). If β is not a constant and M is compact, a detailed study of the critical

7

points of β permits to show that no compact symplectic surface of positive genus admits a non-locallysymmetric preferred symplectic connection. The case of the 2-sphere is more delicate and requiresprecise estimates. The conclusion is

Theorem 3.2 [13] A preferred connection on a compact symplectic surface is necessarily locally sym-metric.

Globally symmetric symplectic surfaces may be completely described. Locally symmetric symplecticsurfaces may be simply related to the globally symmetric ones provided the connection is geodesicallycomplete. A case by case analysis leads to

Theorem 3.3 [13, 4] Let (M,ω,∇) be a compact symplectic surface endowed with a complete locallysymmetric symplectic connection. Then, up to diffeomorphisms, either

• M is the 2-sphere S2 with ω a multiple of the standard volume form and with ∇ the Levi Civitaconnection of the standard Riemannian metric with constant positive curvature equal to 1;

• M is the torus T 2 with ω a multiple of the standard invariant volume form and with ∇ a flataffine symplectic connection;

• M is a surface Σg of genus g ≥ 2 with ∇ a multiple of the standard volume form inherited fromthe disk and with ∇ the connection associated to a metric h of constant negative curvature equalto −1.

What can be said about the existence and the number of different complete preferred connectionson a given compact symplectic surface?

Consider the case of S2 with a symplectic structure ω. There exists a positive real number k so that∫S2 ω = k

∫S2 ω0 where ω0 is the standard symplectic structure on S2 defining the same orientation as ω.

Thus the de Rham cohomology classes defined by ω and kω0 are the same. Furthermore (1− t)ω + tkω0

is symplectic for any t ∈ [ 0 , 1 ] and defines the same cohomology class. By Moser’s stability theorem(see [34]), there exists a 1-parametric family ϕt of diffeomorphisms of the sphere S2 such that ϕ0 = idand ϕ∗1ω = kω0. If g0 is the standard metric on S2, define the metric g on the sphere to be such thatϕ∗1g = kg0. The Levi Civita connection associated to g is clearly symplectic relative to ω and symmetric,hence preferred. Furthermore, if ∇′ is another symplectic connection on (S2, ω) which is preferred andcomplete, it is automatically symmetric since S2 is simply connected and it coincides with ∇. Thus

Theorem 3.4 [13] On (S2, ω) there exists a complete symplectic preferred connection and any completepreferred connection is the image of that one through a symplectomorphism.

The result on the existence is obtained similarly on the torus T 2 or the surface Σg endowed with anysymplectic structure. In these cases, there is no unicity (because of the choice of the affine connectionor the choice of the metric).

The non-compact situation is more complicated. One can show that on the plane endowed withthe standard constant symplectic structure (R2,Ω0), there exist five affinely distinct globally symmetriccomplete symplectic connections. There also exist a 2-parametric family of non-homogeneous preferredsymplectic connections which are not locally symmetric.

8

3.2 Ricci-flat connections

A symplectic connection is said to be Ricci-flat if its Ricci tensor r∇ vanishes. Obviously, those giveexamples of preferred connections! We explain in Section 7 a construction of Ricci-flat connections:when a symplectic manifold (M,ω) of dimension 2n ≥ 4 is the first element of a contact quadruple,any symplectic connection ∇ on (M,ω) can be lifted to define a Ricci-flat symplectic connection on acertain symplectic manifold (P, ω′) of dimension 2n + 2. This procedure gives examples of Ricci-flat,non-flat, symplectic connections in any dimension ≥ 6.

3.3 Ricci-parallel symplectic connections

Let (M,ω,∇) be a symplectic manifold with a symplectic connection; assume there exists on thismanifold a compatible almost complex structure J (i.e. at each point x in M , Jx ∈ End(TxM) , J2

x =− Id , ω(JX, JY ) = ω(X, Y )) which is parallel (i.e. ∇J = 0). Then the pseudo-riemannian metric g

defined by g(X, Y ) := ω(X, JY ) is also parallel, so the connection is the Levi Civita connection for g

and the manifold is pseudo-Kahler. If this connection is preferred one has:

Theorem 3.5 [19] Let (M,ω, J) be a pseudo-Kahler manifold. If the Levi Civita connection is pre-ferred, then the Ricci tensor is parallel.

This gives examples of Ricci-parallel symplectic connections.Consider now a symplectic manifold (M,ω) endowed with a Ricci-parallel symplectic connection ∇.

WriteTxMC = ⊕λ∈spec(TxMC)λ

where (TxMC)λ is the generalised eigenspace for ρ∇x ,i.e.

(TxMC)λ = X ∈ TxMC | (ρ∇x − λ)2nX = 0

and spec denotes the set of different eigenvalues of ρ∇x on TxMC.Observe that (TxMC)λ and (TxMC)µ are orthogonal with respect to the symplectic form ωx unless

µ = −λ. We define real symplectic subspaces of TxM :Rλ, for each real positive eigenvalue λ, so that the complexification of Rλ is given by RC

λ = (TxMC)λ⊕(TxMC)−λ; we denote by R the set of such eigenvalues;Iλ for each purely imaginary eigenvalue λ = ia, a > 0, so that its complexification is IC

λ = (TxMC)λ ⊕(TxMC)−λ; we denote by I the set of such eigenvalues;Cλ for each eigenvalue λ = a + ib, a > 0, b > 0, so that its complexification is CC

λ = (TxMC)λ ⊕(TxMC)−λ ⊕ (TxMC)λ ⊕ (TxMC)−λ; we denote by C the set of such eigenvalues.They give a symplectic orthogonal decomposition of TxM :

TxM = TxM0 ⊕ (⊕λ∈RRλ)⊕ (⊕λ∈IIλ)⊕ (⊕λ∈CCλ).

Ricci-parallel implies that the Ricci endomorphism ρ∇ commutes with all curvature endomorphismsso that all distributions corresponding to the above defined subspaces are parallel. We get

Theorem 3.6 [19, 16] Let (M,ω) be a 2n-dimensional symplectic manifold endowed with a symplecticconnection ∇ whose Ricci tensor is parallel. Assume that the Ricci tensor is not degenerate. Then

9

1 the connection ∇ is the Levi Civita connection associated to the metric defined by the Ricci tensorr∇;

2 the distributions Rλ, Iλ, Cλ are parallel, symplectic, and ρ∇ restricted to any of these is semisimple;

3 if M is simply connected and ∇ complete, then M is symplectomorphic and affinely equivalent tothe product of the symplectic submanifolds corresponding to the integral leaves of the distributionsRλ, Iλ, Cλ;

4 the manifolds corresponding to the leaves of Rλ admit two parallel, transverse Lagrangian folia-tions; those corresponding to Iλ are Kahler-Einstein manifolds; those corresponding to Cλ havedimension 4k if k is the multiplicity of λ and also admit two parallel transverse Lagrangian folia-tions;

5 if all the eigenvalues of ρ∇ have multiplicity one, the factors are two dimensional or four dimen-sional symplectic symmetric spaces.

3.4 Homogeneous preferred connections

If (M,ω) is a connected, simply connected, compact, homogeneous, symplectic manifold, a classicalresult of Kostant tells us that (M,ω) is symplectomorphic to a coadjoint orbit of a compact semi simpleLie group G, endowed with its standard Lie-Kirillov-Kostant-Souriau symplectic structure.

Let ∇ be a symplectic connection on (M,ω) stable by the action of G. Let p ∈ M and let π :G → M g 7→ g.p be the canonical projection related to the choice of the base point p. Let H bethe stabilizer of p in G (H := g ∈ G, | g.p = p ) and let g (resp. h) be the Lie algebra of G (resp.H). For any element X ∈ g, let X∗ denotes the fundamental vector field on M associated to X, i.e.X∗

x = ddt|0 exp−tX.x. Write g = h⊕m where m is the orthogonal to h relative to the Killing form. Each

tangent vector at p is the value at p of a (unique) fundamental vector field Y ∗ with Y ∈ m. Define amap D : m→ End(m) by

(∇X∗Y ∗)p = (D(X)Y )∗p.

This map D entirely determines the G-invariant connection ∇. It satisfies two conditions:

D(X)Y −D(Y )X − πm([X, Y ]) = 0Ω(D(Y )X, Z) + Ω(Y, D(Z)X) = 0 (10)

where πm is the projection of g on m relatively to the decomposition g = h ⊕ m, and where Ω =(π∗ωp)|m×m

. The space of invariant homogeneous symplectic connections on M may be identified to thespace of maps D : m → End(m) satisfying the conditions 10. Such a connection is preferred if it is acritical point of the functional

I (∇) =∫

M

Tr(ρ∇)2ωn

n!.

By Palais’s principle, to determine the G-invariant critical points of I , it is sufficient to determine thecritical points of the restriction of I to the space of G-invariant connections [37]. For an invariantconnection, I reduces to Tr(ρ∇)2 Vol(M) and one shows that the Ricci tensor is a polynomial of degree2 on D.

Lemma 3.7 [18] The functional I is a fourth order polynomial on D.

10

Using the structure of semisimple algebras of compact type, one shows that this polynomial is non-negative and that the homogeneous polynomial corresponding to the terms of degree 4 is strictly positiveoutside the origin. In fact one replaces D by D′ = D − 1

2πm ad; the conditions 10 simply express thefact that the 3-form Ω(D′(·)·, ·) is completely symmetric.

Lemma 3.8 [18] If P : CN → R is a non-negative, real valued polynomial of order d, such that thehomogeneous terms of order d are strictly positive outside the origin, then P has a minimum.

From lemmas 3.7 and 3.8 one gets:

Theorem 3.9 [18] Every coadjoint orbit of a compact semi-simple Lie group G admits a preferredinvariant symplectic connection.

In the case of a coadjoint orbit of SU(3), one can prove by direct calculation that this preferredconnection is unique.

4 Ricci-type connections

Ricci-type connections were defined in Definition 2.8; they are symplectic connections whose curvaturetensor is entirely determined by the Ricci tensor, i.e. for which W∇ = 0.

4.1 Some properties of the curvature of a Ricci-type connection

Let (M,ω) be a smooth symplectic manifold of dim 2n (n ≥ 2) and let ∇ be a smooth Ricci-typesymplectic connection. The following results follow directly from the definition (and Bianchi’s secondidentity).

Lemma 4.1 [20] The curvature endomorphism has the form

R∇(X, Y ) = − 12(n + 1)

[−2ω(X, Y )ρ∇ − ρ∇Y ⊗X + ρ∇X ⊗ Y −X ⊗ ρ∇Y + Y ⊗ ρ∇X] (11)

where X denotes the 1-form i(X)ω (for X a vector field on M) and where, as before, ρ∇ is the endo-morphism associated to the Ricci tensor by r∇(U, V ) = ω(U, ρ∇V ).Furthermore:

(i) there exists a vector field U∇ such that

∇Xρ∇ = − 12n + 1

[X ⊗ U∇ + U∇ ⊗X]; (12)

(ii) there exists a function f∇ such that

∇XU∇ = − 2n + 12(n + 1)

(ρ∇)2X + f∇X; (13)

(iii) there exists a real number K∇ such that

tr(ρ∇)2 +4(n + 1)2n + 1

f∇ = K∇. (14)

Corollary 4.2 Any Ricci-type symplectic connection is preferred

The fact that +XY Z

∇Xr∇(Y, Z) = 0 follows immediately from (12).

11

4.2 A construction by reduction

Consider the manifold M = R2n+2 with its standard symplectic structure Ω′.Let A be a non-zero element in the symplectic Lie algebra sp(R2n+2,Ω′).Let ΣA be the closed hypersurface ΣA ⊂ R2n+2 defined by

ΣA = x ∈ R2n+2|Ω′(x,Ax) = 1.

(In order for ΣA to be non-empty we replace, if necessary, A, by −A.)The 1-parameter subgroup exp tA of the symplectic group acts on R2n+2, preserving Ω′ and ΣA;

[the corresponding fundamental vector field A∗ on R2n+2 (defined by A∗x := ddt exp−tAx|0 = −Ax) is

Hamiltonian, i.e. i(A∗)ω = dHA, with HA(x) = 12Ω(x,Ax) and ΣA is a level set of this Hamiltonian].

We shall consider the reduced space Mred := ΣA/exp tA | t ∈ R with the canonical projectionπ : ΣA →Mred. This can always be locally defined as follows.Since the vector field Ax is nowhere 0 on ΣA, for any x0 ∈ ΣA, there exist:

– a neighbourhood Ux0(⊂ ΣA),– a ball Dred ⊂ R2n of radius r0, centred at the origin,– a real interval I = (−ε, ε)– and a diffeomorphism

χ : Dred × I → Ux0 (15)

such that χ(0, 0) = x0 and χ(y, t) = exp−tA(χ(y, 0)). We shall denote

π : Ux0 → Dred π = p1 ⊗ χ−1.

The space Dred is a local version of the Marsden–Weinstein reduction of ΣA around the point x0.If x ∈ ΣA, TxΣA = 〉Ax〈⊥, where 〉v1, . . . , vp〈 denotes the subspace spanned by v1, . . . , vp and ⊥

denotes the orthogonal relative to Ω′; let Hx(⊂ TxΣA) =〉x,Ax〈⊥; then π∗x defines an isomorphismbetween Hx and the tangent space TyDred for y = π(x).

A reduced symplectic form on Dred, ωred, is defined by

ωredy=π(x)(X, Y ) := Ω′x(Xx, Y x) (16)

where Z denotes the horizontal lift of Z ∈ TyDred; i.e. Z ∈Hx and π∗x(Z) = Z.Let ∇ be the standard flat symplectic affine connection on R2n+2. The reduced symplectic

connection ∇red on Dred is defined by

(∇redX Y )y := π∗x

(∇XY − Ω′(AX,Y )x + Ω′(X,Y )Ax) (17)

Proposition 4.3 [2] The manifold (Dred, ωred) is a symplectic manifold and ∇red is a symplecticconnection of Ricci-type on it.

Furthermore, a direct computation shows that the corresponding ρ∇red

, U∇red

and f∇red

are givenby:

ρ∇redX(x) = −2(n + 1)AxX (18)

U∇red

(x) = −2(n + 1)(2n + 1)A2xx (19)

(π∗f∇red

)(x) = 2(n + 1)(2n + 1)Ω′(A2x,Ax) (20)

12

where Akx is the map induced by Ak with values in Hx:

Akx(X) = AkX + Ω′(AkX, x)Ax− Ω′(AkX, Ax)x.

4.3 Local models for Ricci-type connections

We have seen that given a Ricci-type symplectic connection ∇ on a symplectic manifold (M,ω) thecurvature R∇ is entirely determined by ρ∇ (11); its covariant derivative ∇R∇ is thus determined by∇ρ∇ which in turn is determined by the vector field U∇ (12). The second covariant derivative ∇2R∇ isdetermined by ∇U∇ hence by ρ∇ and f∇ (13). Since f∇ satisfies equation (14), all successive covariantderivatives of the curvature tensor are determined by ρ∇, U∇ and K∇.

Corollary 4.4 Let (M,ω) be a smooth symplectic manifold of dimension 2n (n ≥ 2) and let ∇ bea smooth Ricci-type connection. Let p0 ∈ M ; then the curvature R∇

p0and its covariant derivatives

(∇kR∇)p0 (for all k) are determined by (ρ∇x0, U∇

x0,K∇).

Corollary 4.5 Let (M,ω,∇) (resp. (M ′, ω′,∇′)) be two symplectic manifolds of the same dimension2n (n ≥ 2) each of them endowed with a symplectic connection of Ricci-type. Assume that there existsa linear map b : Tx0M → Tx′0

M ′ such that

(i) b∗ω′x′0= ωx0 ,

(ii) bu∇x0= u∇

′

x′0,

(iii) b ρ∇x0 b−1 = ρ∇

′

x′0.

Assume further that K∇ = K∇′ . Then the manifolds are locally affinely symplectically isomorphic,i.e. there exists a normal neighbourhood of x0 (resp. x′0) Ux0 (resp. U ′

x′0) and a symplectic affine

diffeomorphism ϕ : (Ux0 , ω,∇)→ (U ′x′0

, ω′,∇′) such that ϕ(x0) = x′0 and ϕ∗x0 = b.

In case the symplectic manifold and the connection are real analytic, this follows from classical re-sults, see for instance Theorem 7.2 and Corollary 7.3 in Kobayashi-Nomizu Volume 1 [31]. However,connections of Ricci-type are always real analytic, as we shall see in Section 5.

Theorem 4.6 Any symplectic manifold with a Ricci-type connection is locally symplectically affinelyisomorphic to the symplectic manifold with a Ricci-type connection obtained by a local reduction proce-dure around e0 = (1, 0, . . . , 0) from a constraint surface ΣA defined by a second order polynomial HA

for A ∈ sp(R2n+2,Ω′) in the standard symplectic manifold (R2n+2,Ω′) endowed with the standard flatconnection.

Indeed if p ∈ M and if ξ is a symplectic frame at p [i.e. ξ : (R2n,Ω(2n)) → (Tp, ωp) is a symplecticisomorphism of vector spaces], one defines

u(ξ) = (ξ)−1 U∇(p), ρ(ξ) = (ξ)−1 ρ∇(p) ξ (21)

and

A(ξ) =

0

f(p)2(n + 1)(2n + 1)

−u(ξ)

2(n + 1)(2n + 1)1 0 0

0−u(ξ)

2(n + 1)(2n + 1)−ρ(ξ)

2(n + 1)

(22)

13

with u := Ω′(u, ·) and one looks at the reduction for this A = A(ξ).

4.4 Global models for Ricci-type connections

Theorem 4.7 [21] If (M,ω,∇) is of Ricci type with M simply connected there exists (P, ωP ) symplecticof dimension 2 higher with a flat connection ∇P so that (M,ω,∇) is obtained from (P, ωP ,∇P ) byreduction.

The manifold P is obtained as the product P = N × R of a contact manifold N and the real lineR. The manifold N is the holonomy bundle over M corresponding to a connection defined on theSp(R2n+2,Ω′)-principal bundle

B′(M) = B(M)×Sp(R2n,Ω) Sp(R2n+2,Ω′)

with projection π′ : B(M)′ → M , where B(M) π→ M is the Sp(R2n,Ω)-principal bundle of symplecticframes over M and where we inject the symplectic group Sp(R2n,Ω) into Sp(R2n+2,Ω′) as the set ofmatrices

j(A) =

(I2 00 A

)A ∈ Sp(R2n,Ω).

The connection 1-form α′ on B′(M) is characterised by the fact that

α′[ξ,1]([Xhor

, 0]) = αξ(Xhor

).

where

αξ(Xhor

) =

0

−ωx(u, X)2(n + 1)(2n + 1)

−ρ(X)(ξ)

2(n + 1)0 0 −X(ξ)

X(ξ)−ρ(X)(ξ)2(n + 1)

0

(23)

where X ∈ TxM with x = π(ξ) and Xhor

is the horizontal lift of X in TξB(M)).The equations satisfied by a Ricci-type connection imply that the curvature 2-form of the connection

1-form α′ is equal to −2A′π′∗ω where A′ is the unique Sp(R2n+2,Ω′)-equivariant extension of A to

B′(M); and this curvature 2-form is invariant by parallel transport (dα′curv(α′) = 0).Thus the holonomy algebra of α′ is of dimension 1. Assume M is simply connected. The holonomybundle of α′ is a circle or a line bundle over M , N

π′→ M . This bundle has a natural contact structureν given by the restriction to N ⊂ B(M)′ of the 1-form − 1

2α′ (viewed as real valued since it is valued ina 1-dimensional algebra). One has dν = π′

∗ω.

The symplectic manifold with connection (P, ωP ,∇P ) is then obtained by an induction procedurethat we shall expose in a more general setting in Section 7.1.

5 Special symplectic connections

The striking rigidity results from Section 4 on Ricci-type connections turn out to be a special case of amuch more general phenomenon. As we saw, a connection of Ricci-type can be obtained by a symplecticreduction of a symplectic vector space with a flat symplectic connection. This implies, for example,that the local moduli space of such connections is finite dimensional.

14

As it turns out, this is merely a special case of a much broader phenomenon. Indeed, there are manymore geometric structures which can be characterised in similar terms. For this, we call a symplecticconnection on (M,ω) with dim M ≥ 4 special symplectic if it belongs to one of the following classes.

(i) Connections of Ricci-type (cf. Section 4)

(ii) Bochner–Kahler and Bochner-bi-Lagrangian connections

If the symplectic form is the Kahler form of a (pseudo-)Kahler metric, then its curvature decom-poses into the Ricci curvature and the Bochner curvature ([12]). If the latter vanishes, then (theLevi-Civita connection of) this metric is called Bochner–Kahler.

Similarly, if the manifold is equipped with a bi-Lagrangian structure, i.e. two complementary La-grangian distributions, then the curvature of a symplectic connection for which both distributionsare parallel decomposes into the Ricci curvature and the Bochner curvature. Such a connectionis called Bochner-bi-Lagrangian if its Bochner curvature vanishes.

For results on Bochner–Kahler and Bochner-bi-Lagrangian connections, see [15] and [30] and thereferences cited therein.

(iii) Connections with special symplectic holonomy

A symplectic connection is said to have special symplectic holonomy if its holonomy is containedin a proper absolutely irreducible subgroup of the symplectic group.

The special symplectic holonomies have been classified in [35] and further investigated in [14],[23], [40], [41], [42].

At first, it may seem unmotivated to collect all these structures in one definition, but we shall provideample justification for doing so. Indeed, there is a beautiful link between special symplectic connectionsand parabolic contact geometry.

For this, consider a simple Lie group G with Lie algebra g. We say that g is 2-gradable, if g containsthe root space of a long root. This is equivalent to saying that there is a decomposition as a gradedvector space

g = g−2 ⊕ g−1 ⊕ g0 ⊕ g1 ⊕ g2, and [gi, gj ] ⊂ gi+j , (24)

with dim g±2 = 1. Indeed, there is a (unique) element Hα0 ∈ [g−2, g2] ⊂ g0 such that gi is the eigenspaceof ad(Hα0) with eigenvalue i = −2, . . . , 2, and any non-zero element of g±2 is a long root vector.

Denote by p := g0 ⊕ g1 ⊕ g2 ≤ g and let P ⊂ G be the corresponding connected Lie subgroup. Itfollows that the homogeneous space C := G/P carries a canonical G-invariant contact structure whichis determined by the AdP-invariant distribution g−1mod p ⊂ g/p ∼= TC . In fact, we may regard C asthe projectivisation of the adjoint orbit of a maximal root vector. That is, we view C ⊂ Po(g) wherePo(V ) denotes the set of oriented lines through 0 of a vector space V , so that Po(V ) is the sphere in V .

Each a ∈ g induces an action field a∗ on C with flow Ta := exp(Ra) ⊂ G which hence preservesthe contact structure on C . Let Ca ⊂ C be the open subset on which a∗ is transversal to the contactdistribution. There is a unique contact form α ∈ Ω1(Ca) determined by the equations that α(a∗) ≡ 1.That is, a∗ is a Reeb vector field of the contact form α.

15

We can cover Ca by open sets U such that the local quotient MU := Tloca \U , i.e. the quotient of U

by a sufficiently small neighbourhood of the identity in Ta, is a manifold. Then MU inherits a canonicalsymplectic structure ω ∈ Ω2(MU ) such that π∗(ω) = dα for the canonical projection π : U →MU .

It is now our aim to construct a connection on MU which is ‘naturally’ associated to the givenstructure. For this, we let G0 ⊂ G be the connected subgroup with Lie algebra g0 ≤ g. Since g0 ≤ p

and hence G0 ⊂ P, it follows that we have a fibration

P/G0 −→ G/G0 −→ C = G/P. (25)

In fact, we may interpret G/G0 := (α, v) ∈ T ∗p C × TpC | p ∈ C , α(Dp) = 0, α(v) = 1, whereD ⊂ TC denotes the contact distribution. Thus, given a ∈ g, then for each p ∈ Ca we may regard thepair (αp, a

∗p) from above as a point in G/G0, i.e., we have a canonical embedding ı : Ca → G/G0.

Let Γa := π−1(ı(Ca)) ⊂ G where π : G → G/G0 is the canonical projection. Then the restrictionπ : Γa → ı(Ca) ∼= Ca becomes a principal G0-bundle.

Consider the Maurer-Cartan form µ := g−1dg ∈ Ω1(G) ⊗ g which we decompose according to (24)as µ =

∑2i=−2 µi with µi ∈ Ω1(G)⊗ gi. Then we can show the following.

Proposition 5.1 [22] Let a ∈ g be such that Ca ⊂ C is non-empty, define the action field a∗ ∈ X(C )and the principal G0-bundle π : Γa → Ca with Γa ⊂ G from above. Then we have the following.

(i) The restriction of the components µ0 + µ−1 + µ−2 of the Maurer-Cartan form to Γa yields apointwise linear isomorphism TΓa → g0 ⊕ g−1 ⊕ g−2.

(ii) There is a linear map R : g0 → Λ2(g1)∗⊗g0 and a smooth function ρ : Γa → g0 with the followingproperty. If we define the differential forms κ ∈ Ω1(Γa), θ ∈ Ω1(Γa)⊗ g1 and η ∈ Ω1(Γa)⊗ g0 bythe equation

µ0 + µ−1 + µ−2 = −2κ

(12e−2 + ρ

)+ θ + η

for a fixed element 0 6= e−2 ∈ g−2, then the following equations hold:

dκ =12

< e−2, [θ, θ] >, (26)

anddθ + η ∧ θ = 0,

dη + 12 [η, η] = Rρ(θ ∧ θ).

(27)

Since the Maurer-Cartan form and hence κ, θ and η are invariant under the left action of thesubgroup Ta ⊂ G, we immediately get the following

Corollary 5.2 [22] On Ta\Γa, there is a coframing η + θ ∈ Ω1(Ta\Γa) ⊗ (g0 ⊕ g1) satisfying thestructure equations (27) for a suitable function ρ : Ta\Γa → g0.

Thus, we could, in principle, regard θ and η as the tautological and the connection 1-form, respec-tively, of a connection on the principal bundle Ta\Γa → Ta\Γa/G0 whose curvature is represented byRρ. However, Ta\Γa/G0

∼= Ta\Ca will in general be neither Hausdorff nor locally Euclidean, so thenotion of a principal bundle cannot be defined globally.

16

The way out of this difficulty is to consider local quotients only, i.e., we restrict to sufficiently smallopen subsets U ⊂ Ca for which the local quotient T loc

a \U is a manifold. Clearly, Ca can be covered bysuch open cells.

Moreover, if we describe explicitly the curvature endomorphisms Rρ for ρ ∈ g0, then one can showthat – depending on the choice of the 2-gradable simple Lie algebra g – the connections constructedabove satisfy one of the conditions for a special symplectic connection mentioned before.

More precisely, we have the following

Theorem 5.3 [22] Let g be a simple 2-gradable Lie algebra with dim g ≥ 14, and let C ⊂ Po(g) bethe projectivisation of the adjoint orbit of a maximal root vector. Let a ∈ g be such that Ca ⊂ C isnon-empty, and let Ta = exp(Ra) ⊂ G. If for an open subset U ⊂ Ca the local quotient MU = Tloc

a \Uis a manifold, then MU carries a special symplectic connection.

The dimension restriction on g guarantees that dim MU ≥ 4 and rules out the Lie algebras of typeA1, A2 and B2.

The type of special symplectic connection on MU is determined by the Lie algebra g. In fact, thereis a one-to-one correspondence between the various conditions for special symplectic connections andsimple 2-gradable Lie algebras. More specifically, if the Lie algebra g is of type An, then the connectionsin Theorem 5.3 are Bochner–Kahler of signature (p, q) if g = su(p + 1, q + 1) or Bochner-bi-Lagrangianif g = sl(n, R); if g is of type Cn, then g = sp(n, R) and these connections are of Ricci-type; if g is a2-gradable Lie algebra of one of the remaining types, then the holonomy of MU is contained in one ofthe special symplectic holonomy groups. Also, for two elements a, a′ ∈ g for which Ca,Ca′ ⊂ C are non-empty, the corresponding connections from Theorem 5.3 are equivalent if and only if a′ is G-conjugateto a positive multiple of a.

If Ta∼= S1 then Ta\Ca is an orbifold which carries a special symplectic orbifold connection by

Theorem 5.3. Hence it may be viewed as the “standard orbifold model” for (the adjoint orbit of) a ∈ g.For example, in the case of positive definite Bochner–Kahler metrics, we have C ∼= S2n+1, and forconnections of Ricci-type, we have C ∼= RP2n+1. Thus, in both cases the orbifolds Ta\C are weightedprojective spaces if Ta

∼= S1, hence the standard orbifold models Ta\Ca ⊂ Ta\C are open subsets ofweighted projective spaces.

Surprisingly, the connections from Theorem 5.3 exhaust all special symplectic connections, at leastlocally. Namely we have the following

Theorem 5.4 [22] Let (M,ω) be a symplectic manifold with a special symplectic connection of classC4, and let g be the Lie algebra associated to the special symplectic condition as above.

(i) Then there is a principal T-bundle M →M , where T is a one dimensional Lie group which is notnecessarily connected, and this bundle carries a principal connection with curvature ω.

(ii) Let T ⊂ T be the identity component. Then there is an a ∈ g such that T ∼= Ta ⊂ G, and aTa-equivariant local diffeomorphism ı : M → Ca which for each sufficiently small open subsetV ⊂ M induces a connection preserving diffeomorphism ı : Tloc\V → Tloc

a \U = MU , whereU := ı(V ) ⊂ Ca and MU carries the connection from Theorem 5.3.

The situation in Theorem 5.4 can be illustrated by the following commutative diagram, where thevertical maps are quotients by the indicated Lie groups, and T\M →M is a regular covering.

17

M

T

ı //

T

Ca

Ta

M T\M ı //oo Ta\Ca

(28)

In fact, one might be tempted to summarize Theorems 5.3 and 5.4 by saying that for each a ∈ g, thequotient Ta\Ca carries a canonical special symplectic connection, and the map ı : T\M → Ta\Ca is aconnection preserving local diffeomorphism. If Ta\Ca is a manifold or an orbifold, then this is indeedcorrect. In general, however, Ta\Ca may be neither Hausdorff nor locally Euclidean, hence one has toformulate these results more carefully.

As consequences, we obtain the following

Corollary 5.5 All special symplectic connections of C4-regularity are analytic, and the local modulispace of these connections is finite dimensional, in the sense that the germ of the connection at one pointup to 3rd order determines the connection entirely. In fact, the generic special symplectic connectionassociated to the Lie algebra g depends on (rk(g)− 1) parameters.

Moreover, the Lie algebra s of affine vector fields, i.e., vector fields on M whose flow preserves theconnection, is isomorphic to stab(a)/(Ra) with a ∈ g from Theorem 5.4, where stab(a) = x ∈ g |[x, a] = 0. In particular, dim s ≥ rk(g)− 1 with equality implying that s is abelian.

When counting the parameters in the above corollary, we regard homothetic special symplecticconnections as equal, i.e. (M,ω,∇) is considered equivalent to (M, et0ω,∇) for all t0 ∈ R.

We can generalize Theorem 5.4 and Corollary 5.5 easily to orbifolds. Indeed, if M is an orbifoldwith a special symplectic connection, then we can write M = T\M where M is a manifold and T is aone dimensional Lie group acting properly and locally freely on M , and there is a local diffeomorphismı : M → Ca with the properties stated in Theorem 5.4.

There is a remarkable similarity between the cones Ci ⊂ gi, i = 1, 2, for the simple Lie algebrasg1 := su(n + 1, 1) and g2 := sp(n, R). Namely, C1 = S2n+1 with the standard CR-structure, and g1

is the Lie algebra of the group SU(n + 1, 1) of CR-isomorphisms of S2n+1 [30]. On the other hand,C2 = RP2n+1, regarded as the lines in R2n+2 with the projectivised action of sp(n + 1, R) on R2n+2.Thus, C1 is the universal cover of C2, so that the local quotients Ta\Ca are related. In fact, we havethe following result.

Proposition 5.6 [38] Consider the action of the 2-gradable Lie algebras g1 := su(n + 1, 1) and g2 :=sp(n + 1, R) on the projectivised orbits C1 and C2, respectively. Then the following are equivalent.

(i) For ai ∈ gi the actions of Tai ⊂ Gi on Ci are conjugate for i = 1, 2,

(ii) ai ∈ u(n + 1) where u(n + 1) ⊂ gi for i = 1, 2 via the two standard embeddings.

This together with the preceding results yields the following

Theorem 5.7 [38]

(i) Let (M,ω,∇) be a symplectic manifold with a connection of Ricci type, and suppose that thecorresponding element A ∈ sp(n+1, R) from Theorem 4.6 is conjugate to an element of u(n+1) ⊂sp(n + 1, R). Then M carries a canonical Bochner–Kahler metric whose Kahler form is given byω.

18

(ii) Conversely, let (M,ω, J) be a Bochner-Kahler metric such that the element a ∈ su(n + 1, 1) fromTheorem 5.4 is conjugate to an element of u(n+1) ⊂ su(n+1, 1). Then (M,ω) carries a canonicalconnection of Ricci-type.

Note that in [15], Bochner–Kahler metrics have been locally classified. In this terminology, theBochner–Kahler metrics in the above theorem are called Bochner–Kahler metrics of type I.

6 Symplectic twistor space and Ricci-type connections

In this section we present a result of Vaisman [46] which shows how the Ricci-type condition on thecurvature of a symplectic connection can be seen as an integrability condition for an associated almostcomplex structure on the total space of a bundle over the symplectic manifold. It is far from clear whatis the geometrical significance of this complex structure. Preliminary studies of its properties have beenmade in the PhD theses of Albuquerque [1] and Stienon [43].

6.1 Compatible almost complex structures

Let (V,Ω) be a finite-dimensional real symplectic vector space. We denote by Sp(V,Ω) the real sym-plectic group of linear transformations g of V which preserve Ω. Any two symplectic vector spaces ofthe same dimension are isomorphic, and Sp(V,Ω) acts freely and transitively on the set of isomorphismsfrom (V,Ω) to any other symplectic vector space of the same dimension by composition on the right.

The Lie algebra sp(V,Ω) of Sp(V,Ω) consists of all linear endomorphisms ξ of V satisfying

Ω(ξu, v) + Ω(u, ξv) = 0, ∀u, v ∈ V.

This condition is equivalent to saying that Bξ(u, v) = Ω(u, ξv) defines a symmetric bilinear form Bξ onV . Conversely, any symmetric bilinear form on V defines an element of sp(V,Ω). So as vector spacessp(V,Ω) and S2V ∗ are isomorphic. In fact, if we consider the natural actions of Sp(V,Ω) on both spaceswe have

(g.Bξ)(u, v) = Bξ(g−1u, g−1v) = Ω(g−1u, ξg−1v) = Ω(u, gξg−1v) = BAdgξ(u, v)

so that the adjoint representation of Sp(V,Ω) is isomorphic to the natural representation of Sp(V,Ω)on S2V ∗ ∼= S2V .

A compatible almost complex structure j on (V,Ω) is an element of sp(V,Ω) with j2 = −1 suchthat Bj is positive definite and we denote the set of compatible almost complex structures by j(V,Ω).Sp(V,Ω) acts transitively on j(V,Ω) by conjugation. If we drop the positivity condition, then Sp(V,Ω)has a finite number of orbits distinguished by the signature of Bj . For our purpose any orbit would do,we choose the positive orbit for convenience.

If j is a compatible almost complex structure on (V,Ω) then V becomes a complex vector spaceby defining (a + ib)v = av + bjv,∀a, b ∈ R. Further 〈u, v〉j = Bj(u, v) − iΩ(u, v) defines a positiveHermitean structure on V as a complex vector space. Under the action of Sp(V,Ω) above, it is clearthat the stabiliser of j is the unitary group U(V,Ω, j) of this Hermitean structure so that j(V,Ω) ∼=Sp(V,Ω)/U(V,Ω, j) as homogeneous spaces.

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6.2 Geometry of j(V, Ω)

Since j(V,Ω) is a homogeneous space, the tangent space at j can be identified with the quotient of Liealgebras sp(V,Ω)/u(V,Ω, j). u(V,Ω, j) consists of all elements of sp(V,Ω) which commute with j so theset of elements which anticommute with j form a complementary subspace

mj = ξ ∈ sp(V,Ω) | ξj + jξ = 0.

Since jξj ∈ sp(V,Ω) for ξ ∈ sp(V,Ω) we have a decomposition

ξ =12(ξ − jξj) +

12(ξ + jξj)

with ξ − jξj ∈ u(V,Ω, j) and ξ + jξj ∈ mj . This shows that sp(V,Ω) = u(V,Ω, j) + mj is a direct sumand so we have a natural isomorphism Tjj(V,Ω) ∼= mj .

The product of two endomorphisms which anticommute with j must commute with j so [mj ,mj ] ⊂u(V,Ω, j) and hence Sp(V,Ω)/U(V,Ω, j) is a symmetric space. In fact this is a realisation of the Siegeldomain for the symplectic group. The canonical connection is torsion free and any invariant object isparallel.

If we take ξ ∈ mj then jξ = 12 [j, ξ] ∈ mj and, as j varies, ξ 7→ jξ defines an almost complex

structure on Sp(V,Ω)/U(V,Ω, j) which is clearly Sp(V,Ω)-invariant. Being invariant it is parallel andbeing parallel for a torsion-free connection it is integrable and gives the invariant complex structure ofj(V,Ω) as a Hermitean symmetric space.

If j is an endomorphism of V with j2 = −1 then j extends complex linearly to the complexificationV C and has ±i as eigenvalues with eigenspaces V ±. Then V C = V + + V − is a direct sum and theprojections j± onto these subspaces are given by j± = 1

2 (1∓ ij).Recall that sp(V,Ω) coincides with S2V ∗ ∼= S2V as representations. j acts on both of these as an

element of the Lie algebra, and it is clear that the eigenvalues of j on S2V C will be sums of eigenvaluesof j on V C. Thus they are 0,±2i and the same must hold on sp(V,Ω)C. The zero eigenspace correspondswith u(V,Ω, j)C and the ±2i eigenspaces split mC

j into m+j and m−

j . These are the ±i eigenspaces ofmultiplication by j on the left which defined the complex structure. Thus m+

j is the space of (1, 0)tangent vectors to j(V,Ω) at j. Since there is no 4i eigenspace on sp(V,Ω)C we have [m+

j ,m+j ] = 0 and

since there is no 3i eigenspace on V C, m+j V + = 0. Note that m−

j V + = V − so the condition m+j V + = 0

is a compatibility condition between the complex structures on V and j(V,Ω) which distinguishes thetwo possible invariant complex structures on j(V,Ω) and allows us to choose one in preference to theother.

6.3 The bundle J(M, ω) of almost complex structures

We fix a symplectic vector space (V,Ω) and consider a symplectic manifold (M,ω) of the same dimensionas V . A symplectic frame at x ∈M is a symplectic isomorphism p from (V,Ω) to (TxM,ωx). We denotethe set of symplectic frames at x by Sp(M,ω)x and the disjoint union of these over x ∈M by Sp(M,ω).With the obvious projection map π : Sp(M,ω)→M , Sp(M,ω) is the symplectic frame bundle of (M,ω).It is a principal Sp(V,Ω) bundle with Sp(V,Ω) acting on the right by composition.

Denote by J(M,ω) the bundle on M whose fibre at x is j(TxM,ωx). Smooth sections of J(M,ω)are almost complex structures on M compatible with the symplectic structure so we call J(M,ω) thebundle of compatible almost complex structures. J(M,ω) can also be viewed as the associated bundle

20

Sp(M,ω)×Sp(V,Ω) j(V,Ω) with fibre j(V,Ω). Since the latter is homogeneous, J(M,ω) can be identifiedwith the quotient space Sp(M,ω)/U(V,Ω, j) with the identification given by

Sp(M,ω)/U(V,Ω, j) 3 p.U(V,Ω, j)←→ p j p−1 ∈ J(M,ω). (29)

Sp(M,ω) has a V -valued 1-form θ called the soldering form defined by

θp(X) = p−1(π∗X)

and a vertical bundle V with

Vp = Ker π∗ : TpSp(M,ω)→ Tπ(p)M

so V is the kernel of θ. A 1-form on Sp(M,ω) is called horizontal if it vanishes on V . The componentsof θ span the horizontal forms pointwise.

If ξ ∈ sp(V,Ω) then ξ is the vertical vectorfield on Sp(M,ω) defined by:

ξp =d

dtp exp tξ

∣∣∣∣t=0

.

(p, ξ) 7→ ξp identifies the trivial bundle Sp(M,ω) × sp(V,Ω) with V . Note that the pull-back bundleπ−1TM to Sp(M,ω) is also trivial via the map (p, X) 7→ (p, p−1X) so θ gives a map TSp(M,ω) →Sp(M,ω)× V and an exact sequence of bundles on Sp(M,ω)

0 −→ Sp(M,ω)× sp(V,Ω) −→ TSp(M,ω) −→ Sp(M,ω)× V −→ 0.

Splitting this exact sequence trivialises the tangent bundle of Sp(M,ω) and makes calculations withforms particularly easy.

6.4 Almost symplectic connections

A principal connection in Sp(M,ω) is an almost symplectic connection ∇ in TM (a symplectic connec-tion additionally has vanishing torsion). It is given by an sp(V,Ω)-valued 1-form α on Sp(M,ω) whichsatisfies α(ξ) = ξ and g∗α = Adg α for all ξ ∈ sp(V,Ω) and g ∈ Sp(V,Ω). It follows that H = Ker α isa complementary subbundle to V which we call the horizontal bundle determined by α. ∇ and α arerelated as follows: if U is an open set in M on which there is a local section s of Sp(M,ω) then sv is avectorfield on U for any fixed v in V and s∗α is an sp(V,Ω)-valued 1-form on U which are related by

∇X(sv) = s((s∗α)(X)v)

or, more compactly,s−1∇s = s∗α

with the appropriate interpretation of the two sides of the equation.The components of the forms α, θ are linearly independent and span the cotangent spaces of Sp(M,ω)

at each point so that Sp(M,ω) is parallelisable. This makes the differential geometry on Sp(M,ω)especially simple and is the reason why we shall do calculations on Sp(M,ω) in preference to M orJ(M,ω).

21

The torsion and curvature of ∇ lift to Sp(M,ω) as horizontal V - and sp(V,Ω)-valued 2-forms τ∇

and ρ∇ given by

τ∇p (X, Y ) = p−1(T∇π(p)(π∗X, π∗Y )), ρ∇p (X, Y ) = p−1 R∇π(p)(π∗X, π∗Y ) p.

These forms can be computed from the corresponding principal connection α by

τ∇ = dθ + α ∧ θ, ρ∇ = dα +12[α ∧ α]

where [α ∧ α] denotes simultaneous wedge as forms and Lie bracket of the values.

6.5 Differential geometry on J(M, ω)

Let π be the projection J(M,ω)→M , then π is a submersion so dπ is surjective and we have an exactsequence of vector bundles on J(M,ω)

0 −→ V −→ TJ(M,ω) dπ−→ π−1TM −→ 0

where V now denotes the vertical bundle on J(M,ω). It follows that if j ∈ J(M,ω) then Vj consistsof all elements ξ of sp(TxM,ωx) which anticommute with j where x = π(j). We identify Vj with thissubspace via the map

sp(TxM,ωx) 3 ξ ←→ d

dtexp tξ j exp−tξ

∣∣∣∣t=0

.

End(π−1TM) has a tautological section which we denote by J whose value at j is j ∈ End(Tπ(j)M).If we denote by sp(M,ω) the bundle of Lie algebras whose fibre at x is sp(TxM,ωx) then J is a sectionof π−1sp(M,ω). The kernel of adJ is a bundle u(M,ω, J ) of unitary Lie algebras on J(M,ω). Therange of adJ is the elements of π−1sp(M,ω) which anticommute with J and so coincides with thetangent space to the fibre, namely the vertical bundle V . Thus

π−1sp(M,ω) = u(M,ω, J ) + V .

If ∇ is the almost symplectic connection on TM corresponding to a principal connection α then thehorizontal distribution is U(V,Ω,J )-invariant and hence projects to a horizontal distribution H ∇ onJ(M,ω). Our next objective is to identify H ∇ directly without going via the frame bundle.

The sp(V,Ω) connection form α in Sp(M,ω) pulls back to π∗α in π−1Sp(M,ω) and then pulls backto the reduction as α. It has values in sp(V,Ω) rather than the Lie algebra u(V,Ω, j0) so it is a Cartanconnection from the point of view of this reduction. We can split α relative to the decomposition

sp(V,Ω) = u(V,Ω, j0)⊕mj0

asα = αu + αm.

Obviously, αu is a principal connection in the bundle Sp(M,ω) −→ J(M,ω). αm vanishes on verticalvectors for this bundle so is the lift of a Sp(M,Ω) ×U(V,Ω,j0) mj0-valued 1-form. Since this associatedbundle is V we clearly have a V -valued 1-form on J(M,ω).

Let P be the bundle map TJ(M,ω) −→ π−1sp(M,ω) with image V and kernel H ∇. Then P

anticommutes with J . Note that P can be viewed as a π−1sp(M,ω)-valued 1-form.

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Proposition 6.1 The horizontal lift of P to Sp(M,ω) is αm.

The almost symplectic connection ∇ induces a connection in sp(M,ω) and hence its pull-backπ−1∇ induces one in π−1sp(M,ω). We can thus take the covariant derivative π−1∇J which is also aπ−1sp(M,ω)-valued 1-form. Moreover J 2 = −1 so π−1∇J J + J π−1∇J = 0 and hence π−1∇J

is V -valued. These two 1-forms are related by

Proposition 6.2

π−1∇J = [P,J ].

Proof J lifts to π−1Sp(M,ω) as J (j, p) = pjp−1 and so on the reduction is given by J (σ(p)) =j0. So it is constant. On the reduction the covariant derivative is then σ∗(dJ + [π∗α, J ]) = [α, j0] =[αm, j0]. 2

Finally we can describe the horizontal distribution on J(M,ω) directly.

Corollary 6.3

H ∇ = X ∈ TJ(M,ω) | π−1∇XJ = 0.

6.6 The almost complex structures on J(M, ω)

If (M,ω) is a symplectic manifold the fibres of J(M,ω) are diffeomorphic to j(V,Ω) which has aninvariant complex structure, which then transfers to each fibre. The tangent bundle to the fibres V thushas an endomorphism which gives the almost complex structure of each fibre. Under the identificationof V with endomorphisms of π−1TM this agrees with left multiplication by J .

If ∇ is an almost symplectic connection on (M,ω) then the horizontal distribution H ∇ is isomorphicvia dπ with π−1TM and the latter has an endomorphism J . Since TJ(M,ω) = V ⊕H ∇ there isa unique endomorphism J∇ of TJ(M,ω) which coincides with multiplication by J on V , and whichsatisfies dπJ∇ = J dπ. Clearly J∇ is an almost complex structure on J(M,ω). Thus we have shown

Proposition 6.4 If ∇ is an almost symplectic connection on the symplectic manifold (M,ω) then thereis a unique almost complex structure J∇ on J(M,ω) which satisfies

• PJ∇ = J P, or equivalently, π−1∇J∇XJ = J π−1∇XJ for all X;

• dπJ∇ = J dπ.

It is our goal to prove the following theorem.

Theorem 6.5 Let ∇ be a connection on the symplectic manifold (M,ω).

(i) If ∇ is almost symplectic and J∇ is integrable then there is a symplectic connection ∇′ withJ∇

′= J∇.

(ii) If ∇,∇′ are symplectic connections then J∇′= J∇ implies ∇ = ∇′.

(iii) J∇ is integrable if and only if ∇ is of Ricci-type.

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The first step is to convert the integrability question into a condition on the frame bundle. This ispossible since integrability of an almost complex structure is equivalent to its (1, 0) forms generatinga d-closed ideal. Since the map Sp(M,ω) → J(M,ω) is a submersion, the pull-back of forms is aninjective map, so any relation amongst forms on J(M,ω) will hold if and only if it holds amongst theirpull-backs to Sp(M,ω). Thus we will have integrability if and only if a pointwise basis for the pull-backsof (1, 0) forms generates a d-closed ideal on Sp(M,ω).

Lemma 6.6 The pull-backs to Sp(M,ω) of (1, 0) forms for J∇ are spanned pointwise by α+ = j+αj−

and θ+ = j+θ if α is the connection form for ∇.

See [36] for a proof. An immediate consequence is

Lemma 6.7 J∇ is integrable if and only if j+τ∇ and j+ρ∇j− are in the ideal generated by α+, θ+.

Proof

j+τ∇ − dθ+ = j+α ∧ θ = j+(α(j+ + j−) ∧ θ) = j+α ∧ θ+ + α+ ∧ θ

andj+ρ∇j− − dα+ = j+ρ∇ − dαj− =

12j+[α, α]j− = [αu, α+].

This gives the Lemma immediately. 2

Proof of Theorem 6.5 (i): If ∇ is almost symplectic with torsion T∇ then it is easily verified that theconnection ∇′ defined by

ω(∇′XY, Z) = ω(∇XY, Z)− 12ω(T∇(X, Y ), Z) +

16ω(X, T∇(Y, Z)) +

16ω(Y, T∇(X, Z))

is a symplectic connection. The connection forms α and α′ on Sp(M,ω) are related by

Ω(α′u, v) = Ω(αu, v)− 12Ω(τ∇(θ, u), v) +

16Ω(θ, τ∇(u, v)) +

16Ω(u, τ∇(θ, v)).

Hence

Ω(α′+u, v) = Ω(α′j−u, j−v)

= Ω(α+u, v)− 12Ω(j+τ∇(θ, j−u), v)

+16Ω(θ, j+τ∇(j−u, j−v)) +

16Ω(u, j+τ∇(θ, j−v)).

Integrability implies that j+τ∇(j−u, j−v) = 0 from which one sees that j+τ∇(θ, j−v) = j+τ∇(θ+, j−v)and hence that α′

+ is a combination of α+ and θ+ so the (1, 0) forms of J∇ and J∇′

agree and hencethe almost complex structures are the same.

Proof of Theorem 6.5 (ii): Suppose ∇ and ∇′ are both symplectic connections defining the samealmost complex structures J∇ = J∇

′, which will be the case if and only if the components of α′

+ arecombinations of the components of α+ and θ+. If ∇′ = ∇+ B and we lift B to Sp(M,ω) as

βp(X) = p−1Bπ(p)(π∗X)p

then α′ = α + β so J∇′= J∇ if and only if β+ is a combination of the components of α+ and θ+, and

since β is horizontal, this means β+ vanishes on (0, 1) vectors. Hence j+Bx(j−X)j−Y = 0 for everyj ∈ j(TxM,ωx).

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Solving this condition can be converted to a problem in representation theory by observing first thatit can be re-expressed in terms of the completely symmetric 3-tensor B(X, Y, Z) = ω(B(X)Y, Z) asB(j−X, j−Y, j−Z) = 0. Since the set of such j is Sp(V,Ω)-invariant, this says that B takes its valuesin an invariant subspace in which there is no 3i eigenvalue for any j as an element of the Lie algebra.But S3V is irreducible and the 3i eigenvalue does occur (on S3V +) hence the only invariant subspacewith no 3i eigenvalue is 0. Thus B = 0 and hence ∇′ = ∇.

Proof of Theorem 6.5 (iii): We again write the integrability condition in terms of representation theoryas in the proof of injectivity. The point is that j+ρ∇j− consists of horizontal forms so is in the idealgenerated by (1, 0) forms if and only if j+R∇

x (j−X, j−Y )j−Z = 0 for all j ∈ j(TxM,ωx). This is againan invariant condition, so it means R∇ has to take its values in the largest invariant subspace of thespace of curvature tensors where there is no 4i eigenvalue of j as an element of the Lie algebra. Weknow that there are two irreducible components to curvature, and one is the Ricci component which isisomorphic to symmetric 2-tensors, so only has eigenvalues 0 and ±2i. The other irreducible componentdoes have 4i eigenvalues and so for integrability the curvature must be Ricci-type.

6.7 Generalisations

The fibres of J(M,ω) are large which is the reason the integrability condition kills off so much of thecurvature. We can try to find subbundles with smaller fibres which are complex submanifolds of thefibres of J(M,ω) and then play the same game with this smaller bundle. For example we might have aG-structure P →M for some subgroup G of Sp(V,Ω) and a complex orbit G/H of G on j(V,Ω). ThenP ×G G/H = P/H will have an almost complex structure for each choice of principal G-connectionand a similar calculation can be performed to determine when it is integrable. If G is small thenthe curvature and torsion may break into several irreducible pieces more of which may survive theintegrability condition. On the other hand as G gets smaller it is harder to have such a G-structure

Alternatively, for a given symplectic connection ∇ we can look at the zero-set of the Nijenhuis tensorof J∇ on J(M,ω) for a given symplectic connection ∇. If a component of this set turns out to be acomplex manifold then it can be seen as a twistorial space over M .

Both these approaches lead to interesting examples of twistor spaces in the Riemannian case. Littleis yet known in the symplectic case.

6.8 The bundle J(M, ω) for spaces of Ricci-type

In view of Theorem 6.5 it is interesting to describe the twistor space J(M,ω) for manifolds of Ricci-type in more detail. In particular, since manifolds of Ricci-type can always be obtained locally by thereduction process explained in Theorem 4.6, it seems reasonable to expect the bundle J(M,ω) and itscomplex structure to arise as a reduction of some sort as well. It is the aim of this section to look atthis question.

First of all, consider the twistor space J(R2n+2,Ω′) of R2n+2 with the canonical symplectic structureΩ′, let A ∈ sp(R2n+2,Ω′) and ΣA := x ∈ R2n+2 | Ω′(x,Ax) = 1. We decompose the tangent spacesat x ∈ ΣA ⊂ Rn+2 as

TxΣA = span(Ax)⊕Dx and TxR2n+2 = span(x,Ax)⊕Dx,

25

where Dx is determined by Ω′(span(x,Ax),Dx) = 0. Moreover, we have the canonical projectionπ : ΣA ⊃ U → Mred where U is a sufficiently small open subset such that Mred = U/(exp RA)loc isa manifold, as explained in Section 4.2. Recall that the symplectic structure ω on Mred is determinedby the requirement that dπ|Dx : (Dx,Ω′) → (Tπ(x)M

red, ωπ(x)) becomes a symplectic isomorphism.Moreover, the Ricci-type connection ∇ on (Mred, ω) is defined by

∇XY =(∇0

XY)D

, for all vector fields X, Y on Mred,

where the bar denotes the horizontal lift w.r.t. the distribution D , the subscript D denotes the imageunder the canonical projection TxR2n+2 = span(x,Ax)⊕Dx → Dx, and where ∇0 denotes the canonicalconnection on R2n+2.

Lemma 6.8 Let p : J(Mred, ω) → Mred be the twistor fibration. Then the pull-back of p under themap π : ΣA →Mred is given by the fibration p : ZA → ΣA where

ZA :=(x, J) ∈ J(R2n+2,Ω′) | x ∈ ΣA, Jx = Ax, JAx = −x

, p(x, J) = x

and the map π : ZA → J(Mred, ω) is given by

(π(J))(v) := dπ(Jv), ∀v ∈ Tπ(x)Mred.

Moreover, J(Mred, ω) is the quotient of ZA under the (local) action of exp(RA) given by

exp(tA) · (x, J) := (exp(tA)x, exp(tA) J exp(−tA)).

Proof Let x ∈ ΣA and consider an element J ∈ j(Tπ(x)Mred, ω). Using the symplectic isomorphism

dπx : (Dx,Ω′)→ (Tπ(x)Mred, ω), we see that there is a unique way to extend from Dx to R2n+2 to give

an element J ∈ j(R2n+2,Ω′) with (x, J) ∈ ZA and π(x, J) = J . This shows the first assertion. Thesecond assertion is straightforward. 2

After having described J(Mred, ω) as a manifold as the reduction of ZA under the (local) action ofexp(RA) it would have been nice to see that the twistor almost complex structure on J(Mred, ω) comingfrom the reduced connection arose as a quotient from the twistor complex structure on J(R2n+2,Ω′)coming from ∇0. Unfortunately this is not the case. The inverse image by dπ of the twistor almostcomplex structure on J(Mred, ω) gives an almost complex structure on the quotient of T ZA by RHA andthis would be induced from J∇

0on J(R2n+2,Ω′) only if J∇

0T ZA ⊂ T ZA +RJ∇

0HA. A straightforward

calculation shows that this last condition is equivalent to

[A, J ]Dx = 0, ∀(x, J) ∈ ZA

and that this condition is too strong for general A.

7 Ricci-flat connections

7.1 A construction by induction

Definition 7.1 A contact quadruple (M,N,α, π) is a 2n dimensional smooth manifold M , a 2n + 1dimensional smooth manifold N , a co-oriented contact structure α on N (i.e. α is a 1-form on N suchthat α ∧ (dα)n is nowhere vanishing), and a smooth submersion π : N →M with dα = π∗ω where ω isa symplectic 2-form on M .

26

Definition 7.2 Given a contact quadruple (M,N,α, π) the induced symplectic manifold is the2n + 2 dimensional manifold

P := N × R

endowed with the (exact) symplectic structure

µ := 2e2s ds ∧ p∗1α + e2s dp∗1α = d(e2s p∗1α)

where s denotes the variable along R and p1 : P → N the projection on the first factor.

Induction in the sense of building a (2n+2)-dimensional symplectic manifold from a symplectic manifoldof dimension 2n is also considered by Kostant in [32].

Remark 7.3

• The vector field S := ∂s on P is such that i(S)µ = 2e2s(p∗1α); hence LSµ = 2µ and S is a conformalvector field.• The Reeb vector field Z on N (i.e. the vector field Z on N such that i(Z)dα = 0 and i(Z)α = 1)lifts to a vector field E on P such that: p1∗E = Z and ds(E) = 0. Since i(E)µ = −d(e2s), E is aHamiltonian vector field on (P, µ). Furthermore

[E,S] = 0

µ(E,S) = −2e2s.

• Observe also that if Σ = y ∈ P | s(y) = 0 , the reduction of (P, µ) relative to the constraint manifoldΣ (which is isomorphic to N) is precisely (M,ω).• For y ∈ P define Hy(⊂ TyP ) => E,S <⊥µ . Then Hy is symplectic and (π p1)∗y defines a linearisomorphism between Hy and Tπp1(y)M . Vector fields on M thus admit “horizontal” lifts to P .

We shall now prove that any symplectic connection ∇ on (M,ω) can be lifted to a symplectic connectionon (P, ωP ) which is Ricci-flat. We shall initially define a connection ∇P on P induced by ∇.

First some notation:p denotes the projection p = π p1 : P →M .If X is a vector field on M , ¯X is the vector field on P such that

(i) p∗¯X = X (ii) (p∗1α)( ¯X) = 0 (iii) ds( ¯X) = 0.

Recall that E is the vector field on P such that

(i) p1∗E = Z (ii) ds(E) = 0.

Clearly the values at any point of P of the vector fields ¯X,E, S = ∂s span the tangent space to P atthat point and we have

[E, ∂s] = 0 [E, ¯X] = 0 [∂s,¯X] = 0 [ ¯X, ¯Y ] = [X, Y ]− p∗ω(X, Y )E.

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The formulas for ∇P are:

∇P¯X

¯Y = ∇XY − 12p∗(ω(X, Y ))E − p∗(s(X, Y ))∂s

∇PE

¯X = ∇P¯XE = 2σX + p∗(ω(X, u))∂s

∇P∂s

¯X = ∇P¯X∂s = X

∇PEE = p∗f ∂s − 2U

∇PE∂s = ∇P

∂sE = E

∇P∂s

∂s = ∂s

where f is a function on M , U is a vector field on M , s is a symmetric 2-tensor on M , and σ is theendomorphism of TM associated to s, hence s(X, Y ) = ω(X, σY ).

These formulas have the correct linearity properties and yield a torsion free linear connection on P .One checks readily that ∇P µ = 0 so that ∇P is a symplectic connection on (P, µ).

We now compute the curvature R∇P

of this connection ∇P . We get

R∇P

( ¯X, ¯Y ) ¯Z = R∇(X, Y )Z+2ω(X, Y )σZ − ω(Y, Z)σX + ω(X, Z)σY − s(Y, Z)X + s(X, Z)Y+p∗[ω(X, D(σ,U)(Y, Z))− ω(Y, D(σ,U)(X, Z)]∂s

R∇P

( ¯X, ¯Y )E = 2D(σ,U)(X, Y )− 2D(σ,U)(Y, X)+p∗[ω(X, 1

2fY −∇Y U − 2σ2Y )− ω(Y, 12fX −∇XU − 2σ2X)]∂s

R∇P

( ¯X,E) ¯Y = 2D(σ,U)(X, Y )− p∗[ω(Y, 12fX −∇XU − 2σ2X)]∂s

R∇P

( ¯X,E)E = 2 12fX −∇XU − 2σ2X + p∗[Xf + 4s(X, u)]∂s

R∇P

( ¯X, ¯Y )∂s = 0 R∇P

( ¯X,E)∂s = 0R∇P

( ¯X, ∂s) ¯Y = 0 R∇P

( ¯X, ∂s)E = 0 R∇P

( ¯X, ∂s)∂s = 0R∇P

(E, ∂s) ¯X = 0 R∇P

(E, ∂s)E = 0 R∇P

(E, ∂s)∂s = 0

whereD(σ,U)(Y, Y ′) := (∇Y σ)Y ′ + 1

2ω(Y ′, U)Y − 12ω(Y, Y ′)U.

The Ricci tensor r∇P

of the connection ∇P is given by

r∇P

( ¯X, ¯Y ) = r∇(X, Y ) + 2(n + 1)s(X, Y )r∇

P

( ¯X,E) = −(2n + 1)ω(X, u)− 2 Tr[Y → (∇Y σ)(X)]r∇

P

( ¯X, ∂s) = 0r∇

P

(E,E) = 4Tr(σ2)− 2nf + 2 Tr[X → ∇XU ]r∇

P

(E, ∂s) = 0r∇

P

(∂s, ∂s) = 0

Theorem 7.4 [17] In the framework described above, ∇P is a symplectic connection on (P, µ) for anychoice of s, U and f . The vector field E on P is affine ( LE∇P = 0) and symplectic ( LEµ = 0); thevector field ∂s on P is affine and conformal (L∂sµ = 2µ).

28

Furthermore, choosing

s =−1

2(n + 1)r∇

U : = ω(U, ·) =2

2n + 1Tr[Y → ∇Y σ]

f =1

2n(n + 1)2Tr(ρ∇)2 +

1n

Tr[X → ∇XU ].

we have:

• the connection ∇P on (P, µ) is Ricci-flat (i.e. has zero Ricci tensor);

• if the symplectic connection ∇ on (M,ω) is of Ricci-type, then the connection ∇P on (P, µ) isflat.

• if the connection ∇P is locally symmetric, the connection ∇ is of Ricci-type, hence ∇P is flat.

Proof The first point is an immediate consequences of the formulas above for r∇P

. The second pointis a consequence of the differential identities satisfied by the Ricci-type symplectic connections. Thethird point comes from the fact that (∇P

¯ZR∇P

)( ¯X, ¯Y ) ¯T contains only one term in E whose coefficientis 1

2W∇P

(X, Y, T, Z). 2

7.2 Examples of contact quadruples

We give here examples of contact quadruples corresponding to a given symplectic manifold (M,ω) (i.e.examples of (N,α, π) where N is a smooth 2n + 1 dimensional manifold, α is a 1-form on N such thatα ∧ (dα)n is nowhere vanishing, π : N →M is a smooth submersion and dα = π∗ω.

• Let (M,ω = dλ) be an exact symplectic manifold. Define N = M × R, π = p1 (=projectionof the first factor), α = dt + p∗1λ; then (N,α) is a contact manifold and (M,N,α, π) is a contactquadruple.

The associated induced manifold is P = N × R = M × R2; with coordinates (t, s) on R2 andobvious identification

µ = e2s [ dλ + 2ds ∧ (dt + λ) ].

• Let (M,ω) be a quantizable symplectic manifold; this means that there is a complex linebundle L

p−→M with hermitean structure h and a connection∇ on L preserving h whose curvatureis proportional to iω.Define N := ξ ∈ L | h(ξ, ξ ) = 1 ⊂ L to be the unit circle sub-bundle. It is a principal U(1)bundle and L is the associated bundle L = N ×U(1) C. The connection 1-form on N (representing

∇) is u(1) = iR valued and will be denoted α′; its curvature is dα′ = ikω. Define α :=1ik

α′ andπ := p|N : N →M the surjective submersion. Then (M,N,α, π) is a contact quadruple.

The associated induced manifold P is in bijection with L0 = L\ zero section.

• Let (M,ω) be a connected homogeneous symplectic manifold; i. e. M = G/H where G isa Lie group which we may assume connected and simply connected and where H is the stabilizerin G of a point x0 ∈ M . If p : G → M : g → gx0, p∗ω is a left invariant closed 2-form on G

and Ω = (p∗ω)e, (e=neutral element of G) is a Chevalley 2-cocycle on g (=Lie Algebra of G) with

29

values in R (for the trivial representation).Notice that Ω vanishes as soon as one of its arguments is in h (=Lie algebra of H). Let g1 = g⊕Rbe the central extension of g defined by Ω; i. e.

[(X, a), (Y, b)] = ([X, Y ],Ω(X, Y )).

Let h′ be the subalgebra of g1, isomorphic to h, defined by h′ := (X, 0) |X ∈ h . Let G1 be theconnected and simply connected group of algebra g, and let H ′ be the connected subgroup of G1

with Lie algebra h′. Assume H ′ is closed.Then G1/H ′ admits a natural structure of smooth manifold; define N := G1/H ′. Let p1 : G1 → G

be the homomorphism whose differential is the projection g1 → g on the first factor; clearlyp1(H ′) ⊂ H. Define π : N = G1/H ′ → M = G/H : g1H

′ 7→ p1(g1)H; it is a surjectivesubmersion.

The contact form α on N is constructed as follows: p∗1 p∗ω is a left invariant closed 2-form onG1 vanishing on the fibres of p p1 : G1 → M . Its value Ω1 at the neutral element e1 of G1 isa Chevalley 2-cocycle of g1 with values in R. Define the 1-cochain α1 : g1 → R : (X, a) → −a.Then Ω1 = δα1 is a coboundary. Let α1 be the left invariant 1-form on G1 corresponding to α1.Let q : G1 → G1/H ′ = N be the natural projection.There exists a 1-form α on N so that q∗α = α1. Indeed, for any X ∈ h′ we have

i(X)α1 = α1(X) = 0

(LX α1)((Y, b)) = −α1([X, (Y, b)]) = −α1([X, (Y, b)]) = Ω(X, Y ) = 0

where U is the left invariant vector field on G1 corresponding to U ∈ g1. Furthermore dα = π∗ω

because both are G1 invariant 2-forms on N and:

(dα)q(e1)((X, a)∗N , (Y, b)∗N ) = (q∗dα)e1((X, a), (Y, b))

= (dα1)e1((X, a), (Y, b))

= Ω(X, Y ),

= ωx0(X∗M , Y ∗M )

= (π∗ω)q(e1)((X, a)∗N , (Y, b)∗N )

where we denote by U∗N the fundamental vector field on N associated to U ∈ g1 .

• If (M,ω,∇) is a simply connected symplectic manifold with a Ricci-type connection wehave seen in Section 4.4 how to build the manifold N as a holonomy bundle over M correspondingto a connection built on the extension B′(M) of the frame bundle B(M) over M .

7.3 More about reduction

Let (P, ωP ) be a symplectic manifold of dimension (2n + 2). Assume P admits a conformal vector fieldS:

LSωP = 2ωP ; define α := 12 i(S)ωP so that dα = ωP .

Assume also that P admits a symplectic vector field E commuting with S

LEωP = 0 [S, E] = 0 (⇒ LEα = 0).

30

DefineΣ = x ∈ P |ωP

x (S, E) = 1

and assume that it is non-empty. The tangent space to the hypersurface Σ is given by

TxΣ = ker(i(E)ωP )x = E⊥ωP .

The restriction of ωPx to TxΣ has rank 2n− 2 and a radical spanned by Ex.

The restriction of α to Σ is a contact 1-form on Σ.Let ∼ be the equivalence relation defined on Σ by the flow of E. Assume that the quotient Σ/ ∼

has a 2n dimensional manifold M structure so that π : Σ→ Σ/ ∼= M is a smooth submersion.Define on Σ a “horizontal” distribution of dimension 2n, H , by

H => E, S <⊥ωP ,

and remark that π∗|Hy: Hy → Tx=π(y)M is an isomorphism.

Define as usual the reduced 2-form ωM on M by

ωMx=π(y)(Y1, Y2) = ωP

y (Y1, Y2)

where Yi (i = 1, 2) is defined by (i) π∗Yi = Yi (ii) Yi ∈Hy.Notice that π∗[E, Y ] = 0, and ωP (S, [E, Y ]) = −LEωP (S, Y ) + EωP (S, Y ) = 0 hence

[E, Y ] = 0.

The definition of ωMx does not depend on the choice of y. Indeed

EωP (Y1, Y2) = LEωP (Y1, Y2) + ωP ([E, Y1], Y2) + ωP (Y1, [E, Y2]) = 0.

Clearly ωM is of maximal rank 2n as H is a symplectic subspace. Finally

π∗(dωM (Y1, Y2, Y3)) = +123

(Y1ωM (Y2, Y3)− ωM ([Y1, Y2], Y3))

= +123

(Y1ωP (Y2, Y3)− ωP ([Y1, Y2], Y 3))

and[Y1, Y2] = [Y1, Y2] + ωP (S, [Y1, Y2])E.

Hence ωM is closed and thus symplectic. Clearly π∗ωM = ωP|Σ = d(α|Σ).

Remark 7.5 The manifold M is the first element of a contact quadruple (M,Σ, 12α|Σ , π).

We shall now consider the reduction of a connection. Let (P, ωP ), E, S,Σ,M, ωM be as above. Let∇P be a symplectic connection on P and assume that the vector field E is affine (LE∇P = 0).

Then define a connection ∇Σ on Σ by

∇ΣAB := ∇P

AB − ωP (∇PAB, E)S = ∇P

AB + ωP (B,∇PAE)S.

Then ∇Σ is a torsion free connection and E is an affine vector field for ∇Σ.Define a connection ∇M on M by:

∇MY1

Y2(y) = ∇ΣY1

Y2(y)− ωP (Y2,∇PY1

S)E.

If x ∈ M , this definition does not depend on the choice of y ∈ π−1(x) and one can check that theconnection ∇M is symplectic.

31

Lemma 7.6 [17] Let (P, ωP ) be a symplectic manifold admitting a symplectic connection ∇P , a con-formal vector field S, a symplectic vector field E which is affine and commutes with S. If the constraintmanifold Σ = x ∈ P |ωx(S, E) = 1 is not empty, and if the reduction of Σ is a manifold M , thismanifold admits a symplectic structure ωM and a natural reduced symplectic connection ∇M .

In particular

Theorem 7.7 [17] Let (P, ωP ) be a symplectic manifold admitting a conformal vector field S (LSµ =2µ) which is complete, a symplectic vector field E which commutes with S and assume that, for anyx ∈ P, µx(S, E) > 0. If the reduction of Σ = x ∈ P |µx(S, E) = 1 by the flow of E has a manifoldstructure M with π : Σ→M a surjective submersion, then M admits a reduced symplectic structure ωM

and (P, ωP ) is obtained by induction from (M,ωM ) using the contact quadruple (M,Σ, 12 i(S)ωP

|Σ , π).In particular (P, ωP ) admits a Ricci-flat connection.

8 Non-commutative symplectic symmetric spaces

8.1 Motivations

After the celebrated example of the quantum torus and related non-commutative spaces [39, 24], itappeared natural to try to define non-commutative spaces through oscillatory integral formulae in thesame spirit of [39] but with larger symmetry groups—other than Rd— implementing this way in our classof non-commutative manifolds not only an operator algebraic framework but also a strong geometriccontent.In the context of symmetric spaces, this leads to the so-called ‘WKB-quantisation of symplectic symmet-ric spaces’ as introduced by Karasev, Weinstein and Zakrzewski in the mid 90’s (see [47] and referencestherein). Originally (cf. [47]), the following question was raised in the framework of Hermitean sym-metric spaces of the non-compact type M = G/K.

Question 8.1 Given a G-invariant product on M expressed in the ‘WKB form’:

(u ?θ v)(x) =1

θ2n

∫M×M

aθ(x, y, z) exp(

i

θS(x, y, z)

)u(y) v(z) dy dz , (30)

which conditions on the phase function S ∈ C∞(M3, R) and on the amplitude aθ = a0+θa1+θ2a2+... ∈C∞(M3, R)[[θ]] does one need in order to guarantee formal associativity of ?θ?

Weinstein provided some evidence showing that the phase function S should be closely related to thethree point function denoted hereafter Scan and (partially) defined as follows. Given three points x, y

and z in the symmetric space with the property that the equation sxsysz(X) = X admits a (unique)solution X, the value of Scan(x, y, z) is given by the symplectic area

Scan(x, y, z) =∫

∆XY Z

ω,

where∆

XY Z denotes the geodesic triangle with vertices

X, Y := sz(X), Z = sysz(X).

32

The question of characterising geometrically the amplitude was left open.Another important aspect in this problematic is to determine whether such a WKB product underlies

topological function algebras analogous to the continuous field of C?-algebras deforming C(T) in thecase of the quantum torus Tθ. More precisely:

Question 8.2 Given θ in some deformation parameter space, does one have a function space Aθ

C∞c (M) ⊂ Aθ ⊂ D ′(M)

such that the pair (Aθ, ?θ) is a topological G-algebra1?

In the present section, we survey a geometrical approach to these questions initiated in [5] and based onthe definition of a class of three point functions, called hereafter ‘admissible’, on symplectic symmetricspaces. Admissible functions are characterised by compatibility properties with the symmetries of thesymmetric space at hand. We will show how these properties guarantee associativity of the oscillatingproduct associated to such an admissible function in the case of a cocyclic function. We will then indicateon a curved two dimensional example (a generic coadjoint orbit of the Poincare group in dimension 1+1)how the cocycle condition can be relaxed by introducing a non-trivial amplitude in the oscillating kernel.We will end by mentioning a result which solves Question 8.2 above in this particular context. Let usadd that the general results obtained in this direction go far beyond the particular example presentedhere. For instance, the solution for the general solvable symmetric case has led to several universaldeformation formulae for actions of various classes of solvable Lie groups [10, 9, 6]. Applications in non-commutative geometry (non-commutative causal black holes [7]) as well as in analytic number theory(Rankin–Cohen brackets on modular forms [11]) have been developed.

8.2 Basic definitions and the cocyclic case

Denoting by K := aθ eiθ S the oscillating kernel defining the product given in (30), a simple computation

shows that associativity of ?θ is (at least formally) equivalent to the following condition:∫M

K(a, b, t)K(t, c, d)µ(t) =∫

M

K(a, τ, d)K(τ, b, c)µ(τ), (31)

for every quadruple of points a, b, c, d in M . Equation (31) obviously holds if one can pass from oneintegrand to the other using a change of variables τ = ϕ(t). This motivates

Definition 8.3 Let (M,µ) be an orientable manifold endowed with a volume form µ. A three-pointkernel K ∈ C∞(M ×M ×M) is geometrically associative if for every quadruple of points a, b, c, d

in M there exists a volume preserving diffeomorphism

ϕ : (M,µ)→ (M,µ),

such that for all t in M :

K(a, b, t)K(t, c, d) = K(a, ϕ(t), d)K(ϕ(t), b, c).

1that is, an associative algebra underlying a topological vector space on which the group G acts strongly continuously

by algebra automorphisms.

33

In the sequel, we give sufficient conditions for geometric associativity. From now on, (M,ω, s) denotesa symplectic symmetric space (not necessarily Hermitean). We first observe the following group-likecohomological complex naturally associated to our context.

Definition 8.4 A k-cochain on M is a totally skew symmetric real-valued smooth function S on Mk

which is invariant under the (diagonal) action of the symmetries sxx∈M . Denoting by P k(M), thespace of k-cochains, one has the cohomology operator δ : P k(M)→ P k+1(M) defined as

(δS)(x0, ..., xk) :=∑

j

(−1)jS(x0, ..., xj , ..., xk). (32)

Definition 8.5 A 3-cochain S ∈ P 3(M) is called admissible if for all x ∈M , one has

S(x, y, z) = −S(x, sx(y), z) ∀y, z ∈M.

A Weyl triple is the data of a symmetric space M endowed with an invariant volume form µ togetherwith the data of an admissible 3-cocycle S (i.e. δS = 0).

Skew-symmetry naturally leads us to adopt the following “oriented graph” type notation for a 3-cochainS:

x z

y

• •

•

J

JJJ

def.== S(x, y, z).

A change of orientation in such an “oriented triangle” leads to a change of sign of its value. However,the value represented by such a “triangle” does not depend on the way it “stands”, only the data of thevertices and the orientation of the edges matters.

Now, consider a Weyl triple (M,µ, S), and let A be some (topological) associative algebra. And, forcompactly supported functions u and v ∈ C∞

c (M,A), consider the following “product”:

u ? v(x) =∫

M×M

u(y)v(z)eiS(x,y,z)µ(y) µ(z).

With the above notation for S, associativity for ? now formally takes the following form:

∫M

exp i(

@@

@@

@@

@

-

R

I

•

•

•

•

•

d c

ba

t) µ(t) =

∫M

exp i(

@@

@@

@@

@

6?

I

R

•

•

•

•

•

d c

ba

τ) µ(τ) ,

(33)for every quadruple of points a, b, c, d in M .

In the above formula, the diagram in the argument of the exponential in the LHS (respectively theRHS) stands for S(a, b, t) + S(t, c, d) (respectively S(a, d, τ) + S(τ, b, c)).

34

Proposition 8.6 Let (M,µ, S) be a Weyl triple. Then, the associated three-point kernel K = eiS isgeometrically associative.

Proof. Fix four points a, b, c, d. Regarding Definition 8.3 and formula (33), one needs to construct ourvolume preserving diffeomorphism ϕ : (M,µ)→ (M,µ) in such a way that for all t,

@@

@@

@@

@

-

R

I

•

•

•

•

•

d c

ba

t=

@@

@@

@@

@

6?

I

R

•

•

•

•

•

d c

ba

ϕ(t) .

We first observe that the data of four points a, b, c, d determines what we call an “S-barycentre”, thatis a point g = g(a, b, c, d) such that

@@

@@

@@

@

-

R

I

•

•

•

•

•

d c

ba

g=

@@

@@

@@

@

6?

I

R

•

•

•

•

•

d c

ba

g .

Indeed, since

@@

@@

@@

@

-

R

I

•

•

•

•

•

d c

ba

a—

@@

@@

@@

@

6?

I

R

•

•

•

•

•

d c

ba

a=

a d

c

• •

•

JJ

JJ

—

a c

b

• •

•

JJ

JJ

= —

@@

@@

@@

@

-

R

I

•

•

•

•

•

d c

ba

c+

@@

@@

@@

@

6?

I

R

•

•

•

•

•

d c

ba

c ,

any continuous path joining a to c contains such a point g.

35

Now, we fix once for all such an S-barycentre g for a, b, c, d and we adopt the following notation.For all x and y in M , the value of S(g, x, y) is denoted by a “thickened arrow”:

x g

y

• •

•

J

JJJ

def.== x y• •- .

Again, a change of orientation in such an arrow changes the sign of its value. Also, admissibility whichhas the form

x z

y

• •

•

J

JJJ

=

sx(y) z

x

• •

•

J

JJJ

,

implies

x y• •- = x sg(y)

• •

for all x and y in M . While, from cocyclicity, one gets

x z

y

• •

•

J

JJJ

=

x z

y

• •

•

J

JJJ

.

Moreover, the barycentric property of g can be written

6?

•

•

•

•

d c

ba

=

-

•

•

•

•

d c

ba

.

36

Hence

@@

@@

@@

@

-

R

I

•

•

•

•

•

d c

ba

t=

@@

@@

@@

@

-

R

I

•

•

•

•

•

d c

ba

t=

@@

@@

@@

@

6?

R

I

•

•

•

•

•

d c

ba

t

=

@@

@@

@@

@

6?

I

R

•

•

•

•

•

d c

ba

sg(t)=

@@

@@

@@

@

6?

I

R

•

•

•

•

•

d c

ba

sg(t)

.

One can therefore choose our diffeomorphism ϕ as

ϕ = sg.

Remark 8.7 Given a symplectic symmetric space Weinstein’s function Scan turns out to be admissible(wherever it’s well-defined) [5]. However, the curvature is the obstruction to the cocyclicity of Scan.

8.3 A curved example: SO(1, 1)× R2/R

In what follows, we analyse in some details the case of the solvable symplectic symmetric surfaceM = SO(1, 1)×R2/R. As a homogeneous symplectic manifold it can be realised as a generic coadjointorbit of the Poincare group G = SO(1, 1)×R2. In the dual g? of Lie algebra g of G, the orbit M sits ashyperbolic cylinder. In this picture, the geodesics of the canonical symplectic symmetric connection ∇are planar sections of M ⊂ R3. The affine manifold (M,∇) turns out to be strictly geodesically convex.Moreover, given three points x, y, z in M , the equation sxsysz(t) = t has always a (unique) solutiont ∈M . In particular, Weinstein’s function Scan is everywhere defined on M ×M ×M . Within suitableglobal Darboux coordinates (M,ω) ' (R2, da∧ d`), the symmetry at (a, `) has the following expression:

s(a,`)(a′, `′) = (2a− a′, 2 cosh(a− a′)`− `′) ; (34)

while Weinstein’s function is given by

Scan((a1, `1), (a2, `2), (a3, `3)) = +1,2,3

sinh(a1 − a2)`3 . (35)

Observe that in this coordinate system one sees how far M is from being flat: indeed replacing, in theexpression of the symmetry map, the function

A0 : M ×M → R : ( (a, `) , (a′, `′) ) 7→ cosh(a− a′) (36)

37

by the constant function 1 would yield the flat plane. This function turns out to be exactly the onewhich twists the volume form ω∧ω on M×M in the expression of a WKB quantisation product . Moreprecisely, one has

Theorem 8.8 [5] There exist Frechet function spaces Aθθ∈R such that

(i) for all θ, one hasC∞

c (M) ⊂ Aθ ⊂ C∞(M)

(ii) the formula

(u ?θ v)(x) =1θ2

∫M×M

A0(y, z) exp(

i

θScan(x, y, z)

)u(y) v(z) dy dz

defines an associative product on Aθ θ 6= 0. Each pair (Aθ, ?θ) is then a Frechet algebra.

(iii) For u and v smooth compactly supported functions on M , one has an asymptotic expansion inpowers of θ:

u ?θ v ∼ uv +θ

2iu, v + higher order terms

where , denotes the Poisson structure associated to the symplectic form ω.

An analysis of the G-equivariant (formal) equivalences yields a WKB expression for every star producton M from the data of the preceding one. More precisely, one has

Proposition 8.9 [8] Let P ∈ C∞(R)[[θ]] be a formal function on R. Then, an asymptotic expansionin powers of θ of the following oscillatory integral

(u ?Pθ v)(x) =

1

θ2

ZM×M

P(ax − az)P(ay − ax)

P(ay − az)A0(y, z) exp

„i

θScan(x, y, z)

«u(y) v(z) dy dz (37)

yields a G-invariant star product on M . Moreover, every G-invariant star product on M may bedescribed this way. The choice

P(a) =√

cosh(a) (38)

yields a strongly closed star product ?s.c on M (i.e.∫

u ?s.c v =∫

uv).

Now intrinsically, the amplitude of the oscillating kernel defining the above strongly closed star productmay be described geometrically as follows. Let

Φ : M ×M ×M →M ×M ×M : (x, y, z) 7→ (X, Y, Z) (39)

withsxsysz(X) = X , Y = sz(X) , Z = sy(Y ). (40)

Now consider the Jacobian map of Φ:

JacΦ(x, y, z) :=∣∣∣∣∂(X, Y, Z)

∂(x, y, z)

∣∣∣∣ . (41)

Then, it turns out that in the above coordinate system the function JacΦ depends only on the a-coordinates of the points and that its square root coincides with the above mentioned amplitude. Moreprecisely, one has

38

Theorem 8.10 [8] Let θ > 0. For u and v compactly supported functions on M , the formula

u ?s.c

θ v :=1θ2

∫M×M

√JacΦ e

iθ Scan u⊗ v (42)

extends to L2(M) as an associative product. The function algebra (L2(M), ?s.c

θ ) is then a Hilbert algebrawith respect to the natural Hilbert space structure on L2(M).

From this a continuous field of C?-algebras deforming C0(M) may be obtained via standard techniques(see e.g. [39] for the flat case and [5] for curved solvable symmetric spaces).

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